Key Concept: Force and Interaction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when a football player kicks a ball, the ball moves in the direction of the kick. This is true because applying a force (kick) causes the ball to move. The reason states that a force is a push or pull resulting from the interaction between two objects, which is also true as per the definition of force. Moreover, the reason correctly explains why the ball moves—because the player's foot interacts with the ball, applying a force that sets it in motion.

Key Concept: Balanced Forces

c) The forces are balanced
[Solution Description]
If an object is at rest, it means the net force acting on it is zero \(F_{\text{net}} = 0\), i.e., all forces are balanced. As mentioned in the syllabus: "It means that the forces acting on the object are balancing one another."
This implies no unbalanced force is acting, not that no force is present.

Key Concept: Force, Change in Motion

c) A force acted on the car to change its direction without changing its speed.
[Solution Description]
The question involves understanding the effect of force on an object's motion. Force can change the direction of motion without changing the speed. In this case, since the car's speed remains the same but its direction changes, a force must have acted perpendicular to the initial direction of motion. This force causes the change in direction but does not affect the magnitude of velocity.
Step-by-step reasoning:
1. Initially, the car moves at $10 \, \text{m/s}$ in one direction.
2. After the change, it moves at $10 \, \text{m/s}$ in a perpendicular direction.
3. Since the direction changed, a force must have acted on the car to cause this change, even though the speed remained the same.

Key Concept: Effects of Force

d) Changes speed, direction, or shape of the object
[Solution Description]
According to the syllabus, the application of force can:
1. Make an object move from rest.
2. Change the speed of a moving object.
3. Change the direction of motion of an object.
4. Change the shape of the object.
All these effects are summarized correctly in one of the options.

Key Concept: Definition of Force

b) A push or pull applied on an object
[Solution Description]
A force is defined as a push or pull applied on an object resulting from its interaction with another object. This definition aligns with the basic understanding derived from common experiences like pushing a box or pulling a drawer.

Key Concept: Effect of Force

c) Changing the object's color
[Solution Description]
Forces can cause various effects: making an object move from rest, changing its speed or direction, or altering its shape. However, force does not change the color of an object, as this property is unrelated to mechanical interactions.

Key Concept: Types of Force

c) Opening a drawer
[Solution Description]
Pulling refers to exerting force towards oneself. Among the options, opening a drawer requires pulling it outward, whereas pressing a balloon or kicking a ball involves push forces. Stretching a rubber band also involves pulling.

Key Concept: Effects of Force

c) Changing the direction and speed of the ball
[Solution Description]
When a moving ball is hit by a bat, the force applied changes its direction and may also alter its speed. This is an example of force changing both the direction and speed of an object. The force does not necessarily stop the ball or change its shape in this scenario.

Key Concept: Effects of Force

d) Changing the color of the object.
[Solution Description]
Applying force on an object can cause it to move from rest, change its speed or direction, or alter its shape. However, changing the color of an object is not an effect of force; it usually involves chemical or optical processes. Therefore, this option is correct as it does not align with the known effects of force.

Key Concept: Force effects, Multiple concepts

c) Both speed and direction change gradually
[Solution Description]
Analysis: 1) Initial rolling has kinetic energy (eastward). 2) Kick applies new force vector northward. 3) First effect: eastward motion decreases (force opposes original direction). 4) Second effect: northward motion increases. 5) Net result: direction changes northeast briefly, then north as east component vanishes. This shows multiple force effects simultaneously.

Key Concept: SI Unit of Force

c) Newton
[Solution Description]
The syllabus states that the SI unit of force is newton, represented by the symbol $N$.
Hence, the correct option must specify "newton" as the unit.

Key Concept: Muscular Force

c) Lifting a box from the ground.
[Solution Description]
Muscular force is the force exerted by muscles in the body during physical activities. Among the given options, lifting a box requires contraction and elongation of muscles, making it an example of muscular force. The other options involve forces like gravitational force (apple falling) and contact force (pressing keys), but not necessarily muscular force.

Key Concept: Effects of Force on Objects

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that a force can change the speed of a moving object, which is true as per the syllabus (e.g., hitting a ball changes its speed). The Reason states that applying brakes on a bicycle reduces its speed by applying frictional force, which is also true and directly explains the Assertion because friction (a type of force) opposes motion, reducing speed. Thus, both are true, and the Reason correctly explains the Assertion.

Key Concept: Force, Change in Shape, Application

d) The rod changes shape (either stretches or compresses).
[Solution Description]
When two equal and opposite forces are applied along the length of a metal rod, they create a balanced force scenario. However, depending on the material's elasticity, the rod may experience a change in shape, such as elongation or compression. Since the forces are equal and opposite, the net force is zero, but the rod's shape can still change due to the internal stress caused by the applied forces.

Key Concept: Change in Direction of Motion

c) The direction of the object changes
[Solution Description]
A force applied to a moving object can change its direction. According to the syllabus, one of the effects of force is changing the direction of motion of an object.

Key Concept: Effects of Force

d) It changes the direction of the ball's motion
[Solution Description]
When a bat hits a moving ball, it applies a force that can change the direction of the ball. This demonstrates one of the effects of force—changing the direction of motion of an object.

Key Concept: Effect of Force

c) It changes both the speed and direction of the ball
[Solution Description]
Kicking a moving football applies a force that can change both the speed and direction of the ball depending on how the kick is delivered. However, the primary effect described here is changing the direction of the ball's motion.

Key Concept: Change in Shape Due to Force

b) Pressing an inflated balloon.
[Solution Description]
Forces can change the shape of an object when applied. Pressing an inflated balloon deforms its shape, which matches the given syllabus example where applying push on a balloon changes its shape. The other options involve changes in motion or stopping an object, not shape deformation.

Key Concept: Changing direction and speed

c) Changes both the speed and direction of the car
[Solution Description]
The car's initial velocity ($v_i$) = 20 m/s north. Final velocity ($v_f$) = 10 m/s west. The change involves both a reduction in speed and a change in direction (from north to west). Therefore, the net effect of the force is a combination of changing speed and direction.

Key Concept: Effect of Force

b) It brings the ball to rest
[Solution Description]
When a fielder stops a moving cricket ball by catching it, the force applied opposes the motion of the ball, causing it to slow down and eventually stop. This means the force decreases the speed of the ball.

Key Concept: Effect of Force

b) The ball changes its direction.
[Solution Description]
Hitting a moving ball with a bat applies a force to the ball. This force changes the direction of the ball's motion, as seen in sports like cricket or baseball.

Key Concept: Effect of Force

c) Make the object move from rest.
[Solution Description]
A force applied to a stationary object can make it move from rest. For example, pushing a chair moves it from its initial position.

Key Concept: Force and its Effects

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that applying a force on a moving object always increases its speed. However, this is not always true because a force can also decrease the speed (e.g., brakes on a bicycle), change the direction (e.g., hitting a ball with a bat), or change the shape (e.g., pressing a balloon). The Reason correctly explains the various effects a force can have on an object, but it does not support the Assertion, which is false as it generalizes only one possible effect of force.

Key Concept: Effects of Force on Objects

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when a footballer kicks a stationary football, the force applied causes it to move. This is true because applying a force on a stationary object can set it into motion as per the syllabus (e.g., making an object move from rest). The reason states that a force can make an object move from rest, which is also correct as mentioned in the syllabus. Moreover, the reason correctly explains why the assertion occurs—the force applied by the kick overcomes inertia, causing the football to move from rest. Thus, both are true, and the reason is the correct explanation.

Key Concept: Force and Interaction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a force is a push or pull resulting from interaction, which is true as per the syllabus definition of force. The reason states forces arise only when two objects interact, which is also true and directly explains why forces are interactions between objects. Hence, both are true, and the reason correctly explains the assertion.

Key Concept: Unit of Force

d) Newton, symbol $N$
[Solution Description]
The SI unit of force is newton, named after Sir Isaac Newton, and its symbol is $N$. It is used to measure the magnitude of force.

Key Concept: Force Interactions, Changing Motion

c) The car and wall exert equal and opposite forces on each other while remaining stationary
[Solution Description]
When the car pushes the wall, the wall pushes back with an equal reaction force (Newton's third law). Since the wall doesn't move, these forces cancel out at the interface. The car's wheels push backward on the road, and the road pushes forward on the car with static friction - this is what might eventually make the car move if the force overcomes friction.

Key Concept: Types of Forces

c) Gravitational force
[Solution Description]
Non-contact forces act without direct physical contact. Examples include gravitational force, magnetic force, and electrostatic force, whereas muscular and frictional forces require contact.

Key Concept: Unbalanced forces, Acceleration

b) \(2\,\mathrm{m/s^2}\)
[Solution Description]
First, find the net force acting on the object: 6 N (right) - 2 N (left) = 4 N to the right. Using Newton's Second Law ($F = ma$), we can calculate the acceleration: $a = F/m = 4 N / 2 kg = 2 m/s^2$.

Key Concept: Balanced and Unbalanced Forces

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that an object moving with constant velocity must be experiencing balanced forces. This is true because when forces are balanced, there is no net force acting on the object, resulting in no acceleration (constant velocity or at rest). The reason states that balanced forces result in no change in the state of motion of an object, which is also true and correctly explains why the assertion is true. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Interaction forces, Action-reaction pairs

d) The wall exerts an equal and opposite force back on the person
[Solution Description]
According to Newton's Third Law, when the person exerts a 50 N force on the wall, the wall exerts an equal and opposite force of 50 N back on the person. These forces act on different objects (one on the wall and one on the person) and form an action-reaction pair.

Key Concept: Definition of Force

c) Newton
[Solution Description]
The SI unit of force is newton, represented by the symbol $N$. This is a standard unit defined in physics.

Key Concept: Force Interaction

c) The wall pushes back with equal force
[Solution Description]
According to Newton's third law, every action has an equal and opposite reaction. When the man pushes the wall, the wall exerts an equal and opposite force on the man. Since the wall doesn't move, the reaction force is equal in magnitude to the force applied by the man but in the opposite direction.

Key Concept: Non-Contact Force

c) Gravitational force
[Solution Description]
Non-contact forces act without physical contact between objects. Examples include gravitational force, magnetic force, and electrostatic force. Frictional force requires direct contact between surfaces, making it a contact force.

Key Concept: Force interaction, Balanced forces

d) All forces acting on the car are balanced
[Solution Description]
When an object moves with constant velocity, the net force acting on it is zero. This means all the forces acting on the car are balanced - i.e., the driving force equals the opposing frictional forces.

Key Concept: Force interaction, Newton's third law

d) The book exerts a force of 5 N downward on the table
[Solution Description]
According to Newton's third law, for every action, there is an equal and opposite reaction. Here, the table exerts a force of 5 N upward on the book. Therefore, the book must exert an equal and opposite force of 5 N downward on the table.

Key Concept: Force existence conditions

d) Two neutral particles at rest far apart in space
[Solution Description]
Forces only exist when two objects interact. If there is no physical contact or field interaction (like gravity, electromagnetism), no force exists. A magnet not touching a nail has no contact force, but still experiences magnetic force at a distance. Truly no force exists only when there is no interaction whatsoever.

Key Concept: Force Interaction between Objects

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the concept of forces, whenever two objects interact, each object experiences a force from the other. The assertion states that when you push a table with your hand, both the hand and the table experience a force, which is true. The reason states that a force is always an interaction between two objects, which is also true and correctly explains the assertion. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Force Interaction, Newton's Third Law

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when a book is placed on a table, the table exerts an equal and opposite force on the book to prevent it from falling. This is true because the book exerts a gravitational force (weight) on the table, and the table reacts by exerting a normal force upward, balancing the book's weight. The reason states Newton's Third Law, which explains that forces always occur in pairs (action-reaction pairs). Here, the reason correctly explains the assertion since the interaction between the book and the table follows Newton's Third Law. Thus, both assertion and reason are true, and the reason explains the assertion correctly.

Key Concept: Forces as Interaction Between Two Objects

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that a force can exist without any interaction between two objects, which is false. According to Newton's Third Law of Motion, forces always occur in pairs and result from the interaction between two objects. The Reason correctly explains that forces arise only when two objects interact. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Muscular Force

c) Muscular force
[Solution Description]
When a person lifts a box, the force is generated by the contraction and elongation of muscles in their body. This is called muscular force, which is a type of contact force as it requires direct or indirect physical contact.

Key Concept: Types of Forces

c) Gravitational force
[Solution Description]
Non-contact forces are those that act on objects without physical contact. Examples include gravitational force, magnetic force, and electrostatic force.

Key Concept: Magnetic force, Electrostatic force

d) Both non-contact forces act simultaneously
[Solution Description]
The iron ball experiences magnetic attraction regardless of charge due to its ferromagnetic property. If charged, it also experiences electrostatic force - repulsion if like-charged with the magnet, attraction if oppositely charged. The net force depends on relative strengths, but both non-contact forces are always present.

Key Concept: Contact and Non-Contact Forces

c) Magnetic force
[Solution Description]
Non-contact forces are those that act between objects without any physical contact. Examples include magnetic force, electrostatic force, and gravitational force. Muscular force is a contact force, while frictional force also requires contact.

Key Concept: Contact Forces

c) Muscular force used to lift a bag
[Solution Description]
Muscular force and friction are both contact forces as they require physical interaction between objects. Magnetic force does not require contact.

Key Concept: Muscular Force and Friction

c) The ball slows down and stops due to frictional force
[Solution Description]
The ball gradually slows down and stops due to the force of friction. Friction opposes the motion of the ball when it moves across the surface.

Key Concept: Muscular Force, Friction

b) 53 N
[Solution Description]
First, calculate the frictional force:
$F_{\text{friction}} = \mu \times N$
Here, $N$ is the normal force which equals the weight $W = m \times g$. Assume $g = 9.8 \, \text{m/s}^2$:
$W = 50 \times 9.8 = 490 \, \text{N}$
$F_{\text{friction}} = 0.3 \times 490 = 147 \, \text{N}$
The applied muscular force is 200 N forward, and friction opposes it with 147 N backward:
$F_{\text{net}} = 200 - 147 = 53 \, \text{N}$
So, the net force acting on the sled is 53 N.

Key Concept: Contact Forces

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that muscular force is a contact force, which is true as it requires physical interaction between objects like when we push or pull something using our muscles. The reason explains that muscular force arises from muscle contractions and elongations, which is also correct. Moreover, the reason correctly justifies why muscular force qualifies as a contact force since muscle action involves direct interaction with other objects. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Muscular force, External applications

c) Muscular force, as the oxen contract their muscles to move
[Solution Description]
Oxen use their muscles to generate the force required to pull the plow. Since muscular force is defined as the force resulting from muscle action, the oxen's effort to move the plow is a direct application of muscular force.

Key Concept: Muscular force, Internal body functions

c) Pumping of blood by the heart
[Solution Description]
Muscular force plays a crucial role in internal body functions such as blood circulation. The heart muscles contract and relax rhythmically to pump blood throughout the body, ensuring oxygen and nutrients are delivered to tissues. This process is essential for survival.

Key Concept: Muscular Force

c) It moves food through the gut.
[Solution Description]
During digestion, muscular force helps in chewing food and moving it through the alimentary canal via peristalsis, which involves rhythmic muscle contractions.

Key Concept: Muscular Force

b) The force caused by the action of muscles in our body
[Solution Description]
Muscular force is the force exerted by the contraction and elongation of muscles during physical activities like walking, lifting, or pushing. It is a type of contact force since it occurs when muscles interact with objects or body parts.

Key Concept: Friction, Air Resistance

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that aeroplanes are designed with streamlined shapes to minimize air resistance, which is true. Streamlining reduces the drag force (a type of friction) caused by air molecules colliding with the surface of the aeroplane. However, the reason claims that friction due to air acts in the same direction as the motion of the aeroplane, which is false. Friction always opposes the relative motion between surfaces or objects and thus air resistance acts opposite to the direction of motion of the aeroplane.

Key Concept: Friction, Motion Through Fluids, Conceptual Complexity

b) It doubles
[Solution Description]
Air resistance or drag force in most cases is directly proportional to velocity when the speed is low (viscous drag). If the speed of the object doubles, the air resistance also doubles. For higher speeds, drag may vary with the square of velocity, but the question specifies proportionality to velocity, so the correct interpretation is linear dependence.

Key Concept: Friction and Surface Interaction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that friction reduces the speed of a moving object on a rough surface, which is true because friction opposes motion. The Reason explains why rough surfaces cause more friction by pointing out the increased irregularities leading to greater interlocking, which is also true. Additionally, the Reason correctly explains why the Assertion occurs, as it links the effect (reduced speed) with its cause (greater friction due to surface roughness). Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Friction, Surface Irregularities, Real-World Application

b) 20 N
[Solution Description]
To move the block, the applied force must overcome the static friction. The formula for maximum static friction is $f_s = \mu_s \times N$, where $\mu_s$ is the coefficient of static friction and $N$ is the normal force. Here, the normal force equals the weight of the block, i.e., $N = 50\, \text{N}$. Substituting $\mu_s = 0.4$, we get $f_s = 0.4 \times 50 = 20\, \text{N}$. Thus, the minimum force required to start moving the block is 20 N.

Key Concept: Magnetic force, Gravitational force

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that a magnet can attract an iron nail without touching it, which is true because magnetic force acts at a distance (non-contact). The Reason states that gravitational force is a non-contact force, which is also true as the Earth pulls objects without direct contact. However, the Reason does not explain why the Assertion is correct since gravitational force and magnetic force are two different phenomena. Thus, both statements are true, but the Reason does not correctly explain the Assertion.

Key Concept: Magnetic force, Electrostatic force

a) Both forces originate from the same physical phenomenon but act on different materials
[Solution Description]
Magnetic force acts between magnetic materials or magnets due to alignment of magnetic domains, while electrostatic force arises from the transfer of electrons during friction. Unlike magnetic poles always attract, whereas similarly charged objects always repel. The key difference is their origin: magnetic force is due to dipole interactions, whereas electrostatic force results from excess or deficit of electrons.

Key Concept: Gravitational Force

c) Gravitational force
[Solution Description]
Gravitational force is the non-contact force exerted by the Earth that attracts objects towards its center. This is why a ball thrown upwards eventually falls back to the ground.

Key Concept: Non-contact Forces: Magnetic Force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a magnet can exert force on another magnet without contact, which is true as demonstrated in Activity 5.5. The reason states that magnetic force is a non-contact force, which is also true according to the syllabus. Moreover, the reason correctly explains why the assertion is valid since the non-contact nature of magnetic force allows it to act without physical contact. Thus, both are true, and the reason explains the assertion.

Key Concept: Non-contact force

b) Magnetic force
[Solution Description]
Magnetic force is a non-contact force because it can act over a distance without requiring physical contact between the objects.

Key Concept: Magnetic force, Pole interaction

b) 4 North and 4 South poles
[Solution Description]
Every magnet has a North and South pole, regardless of its size. When a bar magnet is broken, each piece becomes a new magnet with its own North and South poles. Breaking one magnet into four pieces will create four new magnets, each with a North and South pole. Thus, there will be 4 North poles and 4 South poles, totaling 8 poles.

Key Concept: Magnetic force definition

b) The force exerted by a magnet on another magnet or magnetic material
[Solution Description]
The magnetic force is defined as the force exerted by a magnet on another magnet or a magnetic material, even without direct contact. This makes it a non-contact force.

Key Concept: Magnetic force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, when two magnets are brought close to each other, like poles (North-North or South-South) repel each other. This is because the magnetic force between like poles is repulsive. Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Electrostatic force, charge transfer

c) Sphere B becomes negatively charged
[Solution Description]
When the positively charged Sphere A is brought near Sphere B, it induces negative charges on Sphere B's near side and positive charges on its far side. Grounding Sphere B allows the repelled positive charges to escape, leaving Sphere B with a net negative charge after removing the ground. Thus, Sphere B becomes negatively charged.

Key Concept: Electrostatic Force: Charging by Rubbing

c) It transfers electrons from polythene to the straw, charging it negatively.
[Solution Description]
Rubbing two materials together causes friction, which transfers electrons from one material to another. In this case, electrons are transferred from the polythene to the plastic straw, leaving the straw negatively charged and the polythene positively charged. This creates an electrostatic force.

Key Concept: Electrostatic Force Basics

c) Electrostatic force
[Solution Description]
The electrostatic force is the force exerted by a charged body on another charged or uncharged body. It is a non-contact force.

Key Concept: Electrostatic Force: Attraction

b) The paper pieces are attracted to the balloon.
[Solution Description]
When a negatively charged balloon is brought close to uncharged paper pieces, the negative charge on the balloon induces a temporary separation of charges in the paper (polarization). The positive charges in the paper are attracted towards the balloon while the negative charges are repelled. Since the force of attraction between the balloon and the closer positive charges is stronger than the repulsion from the farther negative charges, the paper pieces are pulled towards the balloon.

Key Concept: Gravitational force, Weight measurement

b) 10 N
[Solution Description]
From the given table, we know that an object with mass $1 kg$ weighs $10 N$ on Earth and $25.4 N$ on Jupiter. Since the question states the object's mass is $1 kg$, its weight on Earth would be $10 N$. The weight changes due to different gravitational forces, but mass remains constant.

Key Concept: Weight Unit

b) Newton
[Solution Description]
Weight, being a force, is measured in newtons ($N$). This is part of the International System of Units (SI).

Key Concept: Weight on Different Planets

c) 1.6 N
[Solution Description]
From the syllabus table, the weight of a 1 kg object on the Moon is 1.6 N compared to 10 N on Earth. Therefore, if an object weighs 10 N on Earth, it will weigh approximately $\frac{1.6}{10} \times 10 = 1.6$ N on the Moon.

Key Concept: Non-contact force, Gravitational behavior

c) Earth pulling the moon in its orbit without touching
[Solution Description]
Gravitational force is a non-contact attractive force that acts between two masses without physical contact. Among the options, only option c) correctly describes this property where Earth attracts the moon despite no physical connection.

Key Concept: Weight vs. Mass

c) Newton ($N$)
[Solution Description]
Weight is a force, and its SI unit is newton ($N$), as defined in the syllabus.

Key Concept: Spring balance calibration, weight measurement

c) 0.1 N
[Solution Description]
The spring balance has major marks at every 1 N interval (e.g., 0 N, 1 N, 2 N etc.). Between each major mark there are 10 divisions. To find the smallest measurable weight change:
Weight difference per division = Total difference between major marks / Number of divisions = 1 N / 10 = 0.1 N
Therefore, the smallest measurable weight change is 0.1 N.

Key Concept: Spring Balance

b) 700 g
[Solution Description]
To find the mass of the object:
1. Given that the weight of the object is $7 N$.
2. Use the formula $\text{Mass} = \frac{\text{Weight}}{\text{Gravity}} = \frac{7 N}{10 N/kg} = 0.7 kg = 700 g$.

Key Concept: Difference Between Mass and Weight

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Mass is an intrinsic property of an object and represents the quantity of matter it contains. This value does not depend on the gravitational force acting on the object, so it remains constant regardless of whether the object is on Earth, the Moon, or any other celestial body. On the other hand, weight is the force exerted by gravity on that mass, calculated as $W = m \times g$, where $g$ varies depending on the celestial body. Therefore, the Assertion is true because mass is independent of location. The Reason is also true and correctly explains the Assertion since it states that mass is the amount of matter, which does not change.

Key Concept: Weight and Its Measurement, Weight Measured in Newtons

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Weight is the force exerted by gravity on an object and is given by $W = m \times g$, where $m$ is mass (constant) and $g$ is gravitational acceleration (varies with location). On Earth, $g = 9.8 \, \text{m/s}^2$, while on Mars, $g = 3.7 \, \text{m/s}^2$. Thus, weight changes with location. The reason correctly explains why weight differs due to varying gravitational forces, making both Assertion and Reason true with a valid explanation.

Key Concept: Weight and Its Measurement

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that weight is measured in newtons (N), which is true because weight is a force, and its SI unit is newton. The reason states that weight is the force with which the Earth pulls an object towards itself, which is also true. Moreover, the reason correctly explains why weight is measured in newtons (N) as it defines weight as a force, and force is measured in newtons.

Key Concept: Mass vs. Weight

b) 1 kg
[Solution Description]
From the table, 1 kg mass corresponds to 25.4 N on Jupiter. Therefore, an object weighing 25.4 N has a mass of 1 kg since weight is directly proportional to mass under the same gravitational force.

Key Concept: Spring Balance Measurement

c) Weight of the object in newtons
[Solution Description]
A spring balance measures the gravitational force (weight) acting on an object. The reading directly indicates the weight in newtons (N).

Key Concept: Mass Consistency, Weight Calculation

a) 5 kg, 8 N
[Solution Description]
Mass is constant regardless of location, so the mass remains 5 kg. Weight changes based on gravity. The weight on the Moon is calculated using $Weight = Mass \times Gravitational \ Acceleration$. For the Moon, $Weight = 5 \ kg \times 1.6 \ N/kg = 8 \ N$. Thus, the mass is 5 kg and the weight is 8 N.

Key Concept: Definition of Weight

c) newton ($N$)
[Solution Description]
The SI unit of weight is newton ($N$), as weight is a force measured in the same unit as force.

Key Concept: Weight calculation on different planets

a) 76.2 N
[Solution Description]
The weight on Jupiter is calculated by multiplying the gravitational ratio (2.54) by the weight on Earth. Therefore, $30 \, \text{N} \times 2.54 = 76.2 \, \text{N}$.

Key Concept: Planetary Weight Comparison

a) 6.68
[Solution Description]
The weight on a planet is given by $Weight = Mass \times Gravitational \ Acceleration$. Since mass is constant, the ratio of weights is equal to the ratio of gravitational accelerations. Given weights are 25.4 N (Jupiter) and 3.8 N (Mars). The ratio is $\frac{25.4 \ N}{3.8 \ N} = 6.684$, approximately 6.68 when rounded to two decimal places.

Key Concept: Weight measurement using spring balance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Weight is the force with which a celestial body attracts an object, given by $W = mg$, where $m$ is mass and $g$ is gravitational acceleration. On the Moon, $g$ is about $\frac{1}{6}$ of Earth's $g$. Thus, for the same mass, weight on the Moon is less than on Earth. The assertion states that readings differ on Earth and Moon, which is true as weight depends on $g$. The reason correctly explains this relationship between weight and gravitational acceleration. Hence, both are true, and the reason is the correct explanation.

Key Concept: Measurement of weight using a spring balance

b) 367 g
[Solution Description]
Weight ($W$) is related to mass ($m$) by the formula $W = mg$. Rearranged for mass, $m = \frac{W}{g}$. Substituting the values, $m = \frac{3.6 \, \text{N}}{9.8 \, \text{m/s}^2} \approx 0.367 \, \text{kg}$. To convert kg to grams, multiply by 1000: $0.367 \times 1000 = 367 \, \text{g}$.

Key Concept: Weight vs mass, planetary differences

b) 10 kg
[Solution Description]
Weight = mass × gravity. To find mass, rearrange: $m = \frac{W}{g} = \frac{38 N}{3.8 m/s^2} = 10 kg$.

Key Concept: Mass vs. Weight, Spring Balance Measurement

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the weight of a 1 kg object on Mars is approximately 3.8 N, which is correct as per the given data from the syllabus table (Mars: 1 kg object weighs 3.8 N).
The reason explains that a spring balance measures gravitational force (weight), and the gravitational acceleration on Mars is about 3.8 m/s$^2$. This is also correct because the weight ($W$) of an object is calculated as $W = mg$, where $m$ is mass (constant everywhere) and $g$ is the gravitational acceleration. For Mars, $g \approx 3.8 m/s^2$, so $W = 1 \times 3.8 = 3.8 N$.
The reason correctly explains why the assertion is true since it provides the underlying physics principle linking mass to weight via gravitational acceleration.

Key Concept: Buoyant Force, Archimedes' Principle

c) 20 N
[Solution Description]
First, calculate the buoyant force in water when fully submerged:
Buoyant force = Weight in air - Apparent weight in water = $50 N - 30 N = 20 N$
Since buoyant force equals weight of displaced fluid, for half-submerged case in denser liquid:
New buoyant force = (Original buoyant force / 2) * (Density ratio) = $(20 N / 2) * 2 = 20 N$

Key Concept: Volume Displacement, Multiple Fluids

c) $\rho_s = 0.4\rho_1 + 0.5\rho_2$
[Solution Description]
Using Archimedes' principle for equilibrium:
Weight of sphere = Total buoyant force
$\rho_s V g = \rho_1 (0.4V) g + \rho_2 (0.5V) g$
Cancel $Vg$ from all terms:
$\rho_s = 0.4\rho_1 + 0.5\rho_2$
Remaining 10% is above the interface

Key Concept: Density Comparison, Floating Condition

b) $\rho_A > \rho_w > \rho_C > \rho_B$
[Solution Description]
According to floating conditions:
1. For sinking object: $\rho_A > \rho_w$
2. For partially floating object: $\rho_B = \frac{1}{2}\rho_w$ (since half volume displaces equal weight)
3. For fully floating object: $\rho_C \rho_w > \rho_C > \rho_B$

Key Concept:

b) $w_3 > w_2 > w_1$
[Solution Description]
According to Archimedes’ principle, for floating ($w_1$), the object's weight equals the weight of displaced water. For suspended ($w_2$), the object's weight exactly balances the buoyant force (fully submerged but neither sinks nor rises). For sinking ($w_3$), the object's weight exceeds the buoyant force. Since all have the same volume, the buoyant force is the same when fully submerged. Thus:
$w_1 < w_2 = \text{Buoyant Force} < w_3$

Key Concept: Floating Condition

b) When the weight of displaced liquid is equal to its own weight
[Solution Description]
Archimedes' Principle states that an object floats if the weight of the displaced liquid equals the object's weight. If the buoyant force balances the gravitational force, the object remains afloat.

Key Concept: Buoyant force, Floating condition

b) 20%
[Solution Description]
First, find the volume of the block: $V = \frac{m}{\rho_s} = \frac{2}{800} = 0.0025$ m$^3$.
When floating, the buoyant force equals the block's weight: $\rho_l \cdot V_{\text{submerged}} \cdot g = m \cdot g$.
Substituting values: $1000 \cdot V_{\text{submerged}} = 2$.
Thus, $V_{\text{submerged}} = 0.002$ m$^3$.
The fraction submerged is $\frac{V_{\text{submerged}}}{V} = \frac{0.002}{0.0025} = 0.8$.
Therefore, the fraction above water is $1 - 0.8 = 0.2$ or 20%.

Key Concept: Floating Condition

c) Less than seawater
[Solution Description]
For an object to float, its average density must be less than or equal to the density of the liquid it is placed in. Steel has a higher density than water, but the ship contains empty spaces (e.g., air-filled compartments), reducing its average density below that of seawater.

Key Concept: Buoyant Force and Density

b) 400 kg/m$^3$
[Solution Description]
According to Archimedes' Principle, the buoyant force equals the weight of the displaced liquid. Since the block floats, the weight of the block equals the buoyant force.
Let the volume of the block be $V$ and its density be $\rho_b$. The volume of displaced liquid is $\frac{V}{2}$.
Weight of the block: $W_b = \rho_b \times V \times g$
Weight of displaced liquid: $W_l = \rho_l \times \frac{V}{2} \times g$
Since $W_b = W_l$:
$\rho_b \times V \times g = 800 \times \frac{V}{2} \times g$
Simplifying:
$\rho_b = \frac{800}{2} = 400 \, \text{kg/m}^3$

Key Concept: Multi-concept: Archimedes' Principle, Pressure

c) $2500A(L^2 - 2\frac{H^2}{\cos^2θ})\sinθ$
[Solution Description]
Three forces act: Weight ($W = \rho_r ALg$) downward at $\frac{L}{2}$, Buoyant force $F_b=\rho_w A\left(\frac{H}{\cos\theta}\right)g$ upward at $\frac{H}{2\cos\theta}$, and hinge reaction. Torque due to weight: $\tau_W = W \times \frac{L}{2} \sinθ = 500AL \times 10 \times \frac{L}{2} \sinθ = 2500AL^2 \sinθ$. Torque due to buoyancy: $\tau_b = F_b \times \frac{H}{2\cosθ} \sinθ = 1000A\frac{H}{\cosθ} \times 10 \times \frac{H}{2\cosθ} \sinθ = 5000A \frac{H^2}{\cos^2θ} \sinθ$. Net torque is $\tau_W - \tau_b = 2500AL^2 \sinθ - 5000A \frac{H^2}{\cos^2θ} \sinθ$.

Key Concept: Archimedes' Principle, Buoyant Force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a steel ship floats because its average density is less than water. This is true because ships are designed with hollow structures, reducing their average density below that of water. The reason states that the buoyant force equals the weight of displaced water, which is the correct explanation for why the ship floats (Archimedes' Principle). Thus, both statements are true, and the reason correctly explains the assertion.

Key Concept: Buoyant Force, Gravitational Force

c) Floats with 28% submerged
[Solution Description]
First calculate the shell's volume: $V_{shell} = \frac{4}{3}\pi(0.05^3 - 0.04^3) = 2.56 \times 10^{-4}$ m$^3$. Its mass is $m = \rho V = 7800 \times 2.56 \times 10^{-4} = 2.00$ kg. Weight is $W = mg = 19.6$ N. Volume of mercury displaced if fully submerged equals outer volume: $V_{disp} = \frac{4}{3}\pi(0.05)^3 = 5.24 \times 10^{-4}$ m$^3$. Buoyant force is $F_b = \rho_{Hg} V_{disp} g = 13600 \times 5.24 \times 10^{-4} \times 9.8 = 69.8$ N. Since $F_b > W$, the sphere floats. Fraction submerged: $\frac{W}{F_b} = \frac{19.6}{69.8} = 0.28$.

Key Concept: Density Comparison

c) Density of A is greater than density of B
[Solution Description]
For an object to float, its average density must be less than or equal to the density of water ($1000 \text{ kg/m}^3$). If an object sinks, its density is greater than that of water. Therefore:
$\text{Density of A} > \text{Density of water} > \text{Density of B}.$