NCERT Class 9 Science Solutions: Force and Laws of Motion
Question:
A batsman hits a cricket ball which then rolls on a level ground. After covering a short distance, the ball comes to rest. The ball slows to a stop because ______.
A. the batsman did not hit the ball hard enough.
B. velocity is proportional to the force exerted on the ball.
C. there is a force on the ball opposing the motion.
D. there is no unbalanced force on the ball, so the ball would want to come to rest.
Concept in a Minute:
Newton’s First Law of Motion states that an object in motion will stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force. Friction is a force that opposes motion between two surfaces in contact.
Explanation:
The ball slows down and comes to rest because of the presence of a force that opposes its motion. This force is primarily friction between the ball and the ground, and also air resistance. These forces act in the opposite direction to the ball’s velocity, causing it to decelerate and eventually stop. Option A is incorrect because the initial force from the batsman determines the initial velocity, not the reason it stops. Option B is incorrect; while force causes acceleration (change in velocity), it’s not a direct proportionality to velocity itself in this scenario. Option D is incorrect; if there were no unbalanced force, the ball would continue to roll indefinitely according to Newton’s First Law. Therefore, the correct answer is that there is a force opposing the motion.
Question:
What is the momentum of an object of mass m, moving with a velocity v?
A. (mv) 2
B. mv 2
C. ½ mv 2
D. mv
Concept in a Minute:
Momentum is a fundamental concept in physics that describes the “quantity of motion” an object possesses. It is directly proportional to both the mass of the object and its velocity.
Explanation:
Momentum (p) is defined as the product of an object’s mass (m) and its velocity (v). This relationship is expressed by the formula:
p = mv
Let’s analyze the given options:
A. (mv)² – This represents the square of the momentum, not momentum itself.
B. mv² – This term resembles kinetic energy but with the mass squared, which is incorrect.
C. ½ mv² – This is the formula for kinetic energy, not momentum.
D. mv – This directly matches the definition of momentum.
Therefore, the momentum of an object of mass m, moving with a velocity v, is mv.
Question:
A hammer of mass 500 g, moving at 50 ms−1, strikes a nail. The nail stops the hammer in a very short time of 0.01 s. What is the force of the nail on the hammer?
Concept in a Minute:
This question involves the concept of impulse and momentum. Impulse is the change in momentum of an object. Momentum is the product of mass and velocity. Newton’s second law of motion can also be applied in its rate of change of momentum form.
Explanation:
We are given the mass of the hammer, its initial velocity, and the time it takes to stop. We need to find the force exerted by the nail on the hammer.
Given:
Mass of the hammer (m) = 500 g
Initial velocity of the hammer (u) = 50 m/s
Final velocity of the hammer (v) = 0 m/s (since it stops)
Time taken to stop (Δt) = 0.01 s
First, convert the mass from grams to kilograms:
m = 500 g = 500 / 1000 kg = 0.5 kg
We can calculate the initial momentum of the hammer:
Initial momentum (p_i) = m * u
p_i = 0.5 kg * 50 m/s = 25 kg m/s
The final momentum of the hammer is:
Final momentum (p_f) = m * v
p_f = 0.5 kg * 0 m/s = 0 kg m/s
The change in momentum (Δp) is:
Δp = p_f – p_i
Δp = 0 kg m/s – 25 kg m/s = -25 kg m/s
Impulse is equal to the change in momentum:
Impulse = Δp = -25 kg m/s
Impulse is also defined as the product of force and the time over which the force acts:
Impulse = Force (F) * Δt
Therefore, we can find the force:
F * Δt = Δp
F = Δp / Δt
Substitute the values:
F = -25 kg m/s / 0.01 s
F = -2500 N
The negative sign indicates that the force exerted by the nail on the hammer is in the opposite direction to the hammer’s initial motion. The magnitude of the force of the nail on the hammer is 2500 N.
Alternatively, using Newton’s second law in terms of the rate of change of momentum:
F = (m * v – m * u) / Δt
F = (0.5 kg * 0 m/s – 0.5 kg * 50 m/s) / 0.01 s
F = (0 – 25 kg m/s) / 0.01 s
F = -25 kg m/s / 0.01 s
F = -2500 N
The force of the nail on the hammer is 2500 N, acting in the opposite direction of the hammer’s motion.
Question:
An object of mass 1 kg travelling in a straight line with a velocity of 10 ms−1 collides with, and sticks to, a stationary wooden block of mass 5 kg. Then they both move off together in the same straight line. Calculate the total momentum just before the impact and just after the impact. Also, calculate the velocity of the combined object.
Concept in a Minute:
Conservation of Momentum: In a closed system, the total momentum before a collision is equal to the total momentum after the collision. Momentum is calculated as mass multiplied by velocity (p = mv).
Explanation:
Step 1: Identify the initial state (just before impact).
Calculate the momentum of the object.
Calculate the momentum of the stationary wooden block.
The total momentum just before impact is the sum of these two momenta.
Step 2: Identify the final state (just after impact).
The object and the wooden block stick together, forming a combined object.
The total mass of the combined object is the sum of the individual masses.
Let the velocity of the combined object be v.
The total momentum just after impact is the momentum of the combined object.
Step 3: Apply the Law of Conservation of Momentum.
Equate the total momentum just before impact to the total momentum just after impact.
Step 4: Solve for the velocity of the combined object.
Rearrange the equation from Step 3 to find v.
Given:
Mass of the object (m1) = 1 kg
Initial velocity of the object (u1) = 10 ms⁻¹
Mass of the wooden block (m2) = 5 kg
Initial velocity of the wooden block (u2) = 0 ms⁻¹ (since it is stationary)
Calculation of momentum just before impact:
Momentum of the object (p1) = m1 * u1
Momentum of the wooden block (p2) = m2 * u2
Total momentum before impact = p1 + p2
Calculation of momentum just after impact:
Mass of the combined object (M) = m1 + m2
Let the velocity of the combined object be v.
Momentum of the combined object (P) = M * v
Total momentum after impact = P
Applying Conservation of Momentum:
Total momentum before impact = Total momentum after impact
p1 + p2 = P
Calculating the velocity of the combined object:
Solve the equation M * v = p1 + p2 for v.
Question:
According to the third law of motion, when we push on an object, the object pushes back on us with an equal and opposite force. If the object is a massive truck parked along the roadside, it will probably not move. A student justifies this by answering that the two opposite and equal forces cancel each other. Comment on this logic and explain why the truck does not move.
Concept in a Minute:
Newton’s Third Law of Motion, Inertia, Net Force.
Explanation:
The student’s logic is flawed because it misunderstands how forces interact. While it’s true that the push from the student on the truck and the push from the truck back on the student are equal and opposite (Newton’s Third Law), these forces act on *different* objects. The student’s push acts on the truck, and the truck’s push acts on the student. Forces can only cancel each other out if they act on the *same* object.
The reason the truck does not move is due to inertia and the concept of net force. The truck is a massive object, meaning it has a large amount of inertia. Inertia is the tendency of an object to resist changes in its state of motion. To make the truck move, a net force (an unbalanced force) must be applied to it.
When the student pushes the truck, there are several forces acting on the truck:
1. The student’s pushing force (forward).
2. The frictional force between the truck’s tires and the road (acting backward, resisting motion).
3. The gravitational force pulling the truck down.
4. The normal force from the road pushing the truck up.
For the truck to move, the student’s pushing force must be greater than the opposing frictional force. If the student’s push is not strong enough to overcome the static friction, the net force on the truck will be zero, and it will remain at rest. Even if the student’s push is equal to the static friction, the net force is still zero. The third law’s equal and opposite force acts on the student, not on the truck, and therefore doesn’t cancel the forces acting on the truck. The truck’s immobility is a consequence of the net force acting on it being zero, not because the forces are canceling each other on the truck.
Question:
An object of mass 100 kg is accelerated uniformly from a velocity of 5 ms−1 to 8 ms−1 in 6 s. Calculate the initial and final momentum of the object. Also, find the magnitude of the force exerted on the object.
Concept in a Minute:
Momentum: Momentum is the product of mass and velocity. It is a vector quantity.
Newton’s Second Law of Motion: The rate of change of momentum of an object is directly proportional to the applied force and takes place in the direction of the applied force. Mathematically, F = ma, where F is force, m is mass, and a is acceleration. Also, F = (change in momentum) / (time taken).
Explanation:
The question asks us to calculate the initial and final momentum of an object and the magnitude of the force exerted on it. We are given the mass of the object, its initial velocity, final velocity, and the time taken for this change in velocity.
Step 1: Identify the given information.
Mass of the object (m) = 100 kg
Initial velocity (u) = 5 ms⁻¹
Final velocity (v) = 8 ms⁻¹
Time taken (t) = 6 s
Step 2: Calculate the initial momentum.
Momentum is given by the formula p = mv.
Initial momentum (p₁) = mass × initial velocity = m × u
Step 3: Calculate the final momentum.
Final momentum (p₂) = mass × final velocity = m × v
Step 4: Calculate the change in momentum.
Change in momentum (Δp) = final momentum – initial momentum = p₂ – p₁
Step 5: Calculate the magnitude of the force exerted on the object.
According to Newton’s second law, the force exerted is equal to the rate of change of momentum.
Force (F) = (Change in momentum) / (Time taken) = Δp / t
Step 6: Substitute the given values into the formulas and perform the calculations.
p₁ = 100 kg × 5 ms⁻¹
p₂ = 100 kg × 8 ms⁻¹
Δp = p₂ – p₁
F = Δp / 6 s
Step 7: State the final answers with appropriate units.
The initial momentum, final momentum, and the magnitude of the force will have their respective units (kg ms⁻¹ for momentum and N for force).
Question:
Why is it advised to tie any luggage kept on the roof of a bus with a rope?
Concept in a Minute:
Inertia, specifically inertia of rest and inertia of motion. Inertia is the tendency of an object to resist changes in its state of motion.
Explanation:
When a bus is moving, the luggage on its roof is also moving with the same velocity. If the bus suddenly stops or changes direction, the luggage, due to its inertia, tends to continue in its state of motion. Without being tied, the luggage would continue moving forward, potentially falling off the roof or causing an accident. Tying the luggage with a rope provides an external force that overcomes its inertia and keeps it in place relative to the bus.
To elaborate:
When the bus is moving at a constant speed, the luggage is at rest relative to the bus. This is inertia of rest. If the bus stops abruptly, the luggage’s inertia of rest makes it want to remain at rest, which would mean it stays put relative to the ground while the bus stops. However, since it’s on the roof of the bus, it’s essentially being dragged along by the bus. Without restraint, its inertia would cause it to continue moving forward relative to the bus, which is now stationary.
Conversely, if the bus starts moving from rest, the luggage’s inertia of rest makes it resist this change. It tends to stay at rest, and if not secured, it might slide backward relative to the accelerating bus.
When the bus takes a turn, the luggage experiences inertia of motion. It tends to continue moving in a straight line, while the bus is changing its direction. This lateral force would cause the luggage to slide sideways. Tying it down with a rope exerts a centripetal force that keeps the luggage moving along with the curved path of the bus.
Question:
A hockey ball of mass 200 g travelling at 10 m s−1 is struck by a hockey stick so as to return it along its original path with a velocity at 5 m s−1. Calculate the change of momentum occurred in the motion of the hockey ball by the force applied by the hockey stick.
Concept in a Minute:
Momentum is a measure of mass in motion and is calculated as the product of an object’s mass and its velocity (p = mv). The change in momentum (also called impulse) is the difference between the final momentum and the initial momentum of an object. A change in momentum implies a force has been applied.
Explanation:
First, we need to convert the mass of the hockey ball from grams to kilograms.
Mass (m) = 200 g = 0.2 kg
The initial velocity of the hockey ball is given as 10 m s⁻¹. Let’s consider the original path as the positive direction.
Initial velocity (u) = +10 m s⁻¹
The final velocity is in the opposite direction along the original path with a speed of 5 m s⁻¹. Therefore, the final velocity is negative.
Final velocity (v) = -5 m s⁻¹
Now, we calculate the initial momentum (pᵢ) and the final momentum (p<0xE2><0x82><0x9F>) of the hockey ball.
Initial momentum (pᵢ) = m × u = 0.2 kg × 10 m s⁻¹ = 2 kg m s⁻¹
Final momentum (p<0xE2><0x82><0x9F>) = m × v = 0.2 kg × (-5 m s⁻¹) = -1 kg m s⁻¹
The change of momentum (Δp) is the difference between the final momentum and the initial momentum.
Change of momentum (Δp) = p<0xE2><0x82><0x9F> – pᵢ
Δp = (-1 kg m s⁻¹) – (2 kg m s⁻¹)
Δp = -3 kg m s⁻¹
The negative sign indicates that the change in momentum is in the direction opposite to the initial motion of the ball. The magnitude of the change in momentum is 3 kg m s⁻¹.
Answer:
The change of momentum occurred in the motion of the hockey ball is -3 kg m s⁻¹.
Question:
Which of the following has more inertia:
- a rubber ball and a stone of the same size?
- a bicycle and a train?
- a five-rupees coin and a one-rupee coin?
Concept in a Minute:
Inertia is the resistance of an object to a change in its state of motion. This means that an object at rest tends to stay at rest, and an object in motion tends to stay in motion with the same speed and in the same direction, unless acted upon by an external force. Inertia is directly proportional to the mass of an object. The greater the mass, the greater the inertia.
Explanation:
To determine which object has more inertia, we need to compare their masses. The object with the larger mass will have more inertia.
a) A rubber ball and a stone of the same size:
A stone is generally much denser and therefore has a greater mass than a rubber ball of the same size. Thus, the stone has more inertia.
b) A bicycle and a train:
A train is significantly more massive than a bicycle. Therefore, the train has much more inertia.
c) A five-rupees coin and a one-rupee coin:
While both are coins, a five-rupees coin (in the current Indian currency system) is typically made of different materials or is of a slightly different size and weight than a one-rupee coin, resulting in a difference in mass. Generally, the five-rupees coin is heavier and therefore has more mass and more inertia. (Note: This comparison assumes standard denominations. If the question intended two coins of identical material and size but different face values, then the premise might be flawed. However, for standard denominations, a difference in mass is expected.)
Question:
How much momentum will a dumb-bell of mass 10 kg transfer to the floor if it falls from a height of 80 cm? Take its downward acceleration to be 10 m s−2.
Concept in a Minute:
Momentum is the product of mass and velocity (p = mv). To find the momentum transferred, we need to find the velocity of the dumbbell just before it hits the floor. Since the dumbbell is falling, we can use kinematic equations to find its final velocity. The question asks for the momentum transferred to the floor, which will be equal to the change in momentum of the dumbbell.
Explanation:
The momentum transferred to the floor is equal to the change in momentum of the dumbbell. The dumbbell starts with zero momentum (at rest) and gains momentum as it falls. Just before hitting the floor, its momentum is given by p = mv, where m is the mass and v is the final velocity. We can find the final velocity using a kinematic equation.
Let:
m = mass of the dumbbell = 10 kg
h = height of fall = 80 cm = 0.8 m
a = downward acceleration = 10 m s⁻²
u = initial velocity = 0 m s⁻¹ (starts from rest)
v = final velocity just before hitting the floor
We use the kinematic equation: v² = u² + 2as
v² = 0² + 2 * 10 m s⁻² * 0.8 m
v² = 16 m²/s²
v = √16 m²/s²
v = 4 m s⁻¹
Now, calculate the momentum just before impact:
p = mv
p = 10 kg * 4 m s⁻¹
p = 40 kg m s⁻¹
The momentum transferred to the floor is equal to the change in momentum of the dumbbell. Since the dumbbell starts from rest, the change in momentum is its final momentum.
Therefore, the momentum transferred to the floor is 40 kg m s⁻¹.
Question:
Two persons manage to push a motorcar of mass 1200 kg at a uniform velocity along a level road. The same motorcar can be pushed by three persons to produce an acceleration of 0.2 m s−2. With what force does each person push the motorcar? (Assume that all persons push the motorcar with the same muscular effort.)
Concept in a Minute:
Newton’s second law of motion (F = ma) and the concept of net force. When multiple forces act on an object, the net force is the vector sum of all individual forces. When an object moves at a uniform velocity, the net force acting on it is zero.
Explanation:
Let m be the mass of the motorcar, which is 1200 kg.
Let F be the force exerted by each person.
Case 1: Two persons push the motorcar at a uniform velocity.
When the motorcar moves at a uniform velocity, its acceleration is 0 m/s².
According to Newton’s second law, the net force acting on the car is zero.
The forces acting on the car are the pushing force from the two persons and the opposing frictional force from the road.
Let the pushing force be 2F (since each person exerts the same force F).
Let f be the frictional force.
Net force = Pushing force – Frictional force
0 = 2F – f
Therefore, the frictional force f = 2F.
Case 2: Three persons push the motorcar to produce an acceleration of 0.2 m/s².
The pushing force from three persons is 3F.
The frictional force remains the same, which is f = 2F.
The net force acting on the car is given by Newton’s second law:
Net force = m * a
3F – f = m * a
Substitute the value of f from Case 1:
3F – 2F = m * a
F = m * a
Now, plug in the given values:
m = 1200 kg
a = 0.2 m/s²
F = 1200 kg * 0.2 m/s²
F = 240 N
So, the force with which each person pushes the motorcar is 240 N.
Question:
A 8000 kg engine pulls a train of 5 wagons, each of 2000 kg. along a horizontal track. If the engine exerts a force of 40000 N and the track offers a friction force of 5000 N, then calculate:
- the net accelerating force and
- the acceleration of the train.
Concept in a Minute:
Newton’s Second Law of Motion (F_net = ma), where F_net is the net force acting on an object, m is its mass, and a is its acceleration. Force is a vector quantity, so we need to consider both the applied force and opposing forces like friction.
Explanation:
Part a) the net accelerating force and
The engine exerts a forward force. The friction force acts in the opposite direction, opposing the motion. Therefore, the net accelerating force is the difference between the applied force and the friction force.
Net accelerating force = Force exerted by engine – Friction force
Net accelerating force = 40000 N – 5000 N
Net accelerating force = 35000 N
Part b) the acceleration of the train.
First, we need to calculate the total mass of the train. The train consists of the engine and 5 wagons.
Mass of engine = 8000 kg
Mass of each wagon = 2000 kg
Total mass of 5 wagons = 5 * 2000 kg = 10000 kg
Total mass of the train = Mass of engine + Total mass of wagons
Total mass of the train = 8000 kg + 10000 kg = 18000 kg
Now, we can use Newton’s Second Law of Motion (F_net = ma) to calculate the acceleration. We have already calculated the net accelerating force (F_net) and the total mass (m).
F_net = 35000 N
m = 18000 kg
a = F_net / m
a = 35000 N / 18000 kg
a = 35 / 18 m/s^2
a ≈ 1.94 m/s^2
Question:
Why do you fall in the forward direction when a moving bus brakes to a stop and fall backwards when it accelerates from rest?
Concept in a Minute:
Inertia: Inertia is the tendency of an object to resist changes in its state of motion. This includes resisting changes in its velocity (speed and direction). Newton’s First Law of Motion (Law of Inertia) states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
Explanation:
When a moving bus brakes to a stop, your body, due to inertia, tends to continue moving forward at the same speed the bus was previously traveling. The bus stops because of the braking force applied to it. However, your body is not directly connected to the bus’s braking system, so it continues its forward motion until an external force (like friction with the seat or you grabbing something) stops it. This makes you fall forward relative to the bus.
Conversely, when a bus accelerates from rest, your body, due to inertia, tends to remain at rest. The bus moves forward due to the engine’s force. Your body, however, tries to stay in its original position. This resistance to acceleration makes it feel like you are being pushed backward relative to the accelerating bus. It is the bus that is moving forward, and your body is lagging behind due to inertia.
Question:
Using a horizontal force of 200 N, we intend to move a wooden cabinet across a floor at a constant velocity. What is the friction force that will be exerted on the cabinet?
Concept in a Minute:
Newton’s First Law of Motion (Law of Inertia) states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. For an object to move at a constant velocity, the net force acting on it must be zero. Friction is a force that opposes motion.
Explanation:
The question states that the wooden cabinet is being moved across a floor at a constant velocity. According to Newton’s First Law of Motion, if an object is moving at a constant velocity, the net force acting on it is zero. This means that all the forces acting on the object are balanced.
We are given a horizontal force of 200 N applied to move the cabinet. Since the cabinet is moving at a constant velocity, there must be another horizontal force acting in the opposite direction to counteract the applied force. This opposing force is the friction force.
For the net force to be zero, the applied force and the friction force must be equal in magnitude and opposite in direction. Therefore, if the applied horizontal force is 200 N, the friction force exerted on the cabinet must also be 200 N, acting in the direction opposite to the motion.
Friction Force = Applied Force
Question:
Explain why some of the leaves may get detached from a tree if we vigorously shake its branch.
Concept in a Minute:
Inertia: An object at rest tends to stay at rest, and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
Force: A push or pull that can change the state of motion of an object.
Attachment of leaves: Leaves are attached to the branch by a structure called the petiole, which contains vascular tissues and has a certain strength.
Explanation:
When a tree branch is vigorously shaken, a large force is applied to the branch, causing it to move rapidly. However, the leaves, due to their inertia, tend to remain in their original position. This difference in motion between the branch and the leaves creates a significant tension or stress at the point where the leaf is attached to the branch (the petiole). If the force applied by shaking is strong enough to overcome the strength of the petiole and the forces holding the leaf to the branch, the leaves will be detached and fall. Essentially, the leaf’s resistance to sudden movement causes it to break away from the moving branch.
Question:
An object experiences a net zero external unbalanced force. Is it possible for the object to be travelling with a non-zero velocity? If yes, state the conditions that must be placed on the magnitude and direction of the velocity. If no, provide a reason.
Concept in a Minute:
Newton’s First Law of Motion states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external unbalanced force. This implies that a net zero external unbalanced force means the object will maintain its state of motion. Uniform motion implies constant velocity, which has both magnitude (speed) and direction.
Explanation:
Yes, it is possible for the object to be travelling with a non-zero velocity.
The condition is that the object must be in a state of uniform motion.
This means:
1. Magnitude of velocity: The speed of the object must be constant. It should not be increasing or decreasing.
2. Direction of velocity: The direction of the object’s motion must be constant. It should not be changing.
Therefore, if the object is already moving with a constant speed in a straight line, and the net external unbalanced force acting on it is zero, it will continue to move with that same constant velocity.
Question:
An automobile vehicle has a mass of 1500 kg. What must be the force between the vehicle and road if the vehicle is to be stopped with a negative acceleration of 1.7 m s−2?
Concept in a Minute:
Newton’s second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F=ma).
Explanation:
The question asks for the force required to stop a vehicle with a given mass and a specific negative acceleration.
1. Identify the given information:
– Mass of the vehicle (m) = 1500 kg
– Negative acceleration (a) = -1.7 m s⁻² (The negative sign indicates deceleration, which is what happens when stopping).
2. Recall Newton’s second law of motion:
Force (F) = mass (m) × acceleration (a)
3. Substitute the given values into the formula:
F = 1500 kg × (-1.7 m s⁻²)
4. Calculate the force:
F = -2550 N
The force required between the vehicle and the road is -2550 N. The negative sign indicates that the force is acting in the opposite direction of the vehicle’s motion, which is necessary for it to slow down and stop. Therefore, the magnitude of the force is 2550 N.
Question:
When a carpet is beaten with a stick, dust comes out of it. Explain.
Concept in a Minute:
Inertia of rest is the tendency of an object to remain at rest. According to Newton’s first law of motion, an object at rest stays at rest unless acted upon by an external force.
Explanation:
When a carpet is beaten with a stick, the stick applies a force on the carpet. This force causes the carpet to move. However, the dust particles, due to their inertia of rest, tend to remain in their original position. As the carpet moves away rapidly, the dust particles are left behind, thus appearing to come out of the carpet.
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