FORCE AND LAWS OF MOTION - Q&A

Questions Page 91

1. Which of the following has more inertia:
(a) a rubber ball and a stone of the same size?
(b) a bicycle and a train?
(c) a five-rupees coin and a one-rupee coin?

Inertia is the measure of the mass of an object. The greater the mass, the greater the inertia.
(a) A stone has more inertia because it has more mass than a rubber ball of the same size.
(b) A train has more inertia because it is much heavier than a bicycle.
(c) A five-rupees coin has more inertia because it has more mass than a one-rupee coin.

2. In the following example, try to identify the number of times the velocity of the ball changes:
"A football player kicks a football to another player of his team who kicks the football towards the goal. The goalkeeper of the opposite team collects the football and kicks it towards a player of his own team."
Also identify the agent supplying the force in each case.

The velocity of the ball changes four times:
1. First change: When the first player kicks the stationary football.
Agent: First player.
2. Second change: When the second player kicks the ball towards the goal.
Agent: Second player.
3. Third change: When the goalkeeper collects (stops) the ball.
Agent: Goalkeeper.
4. Fourth change: When the goalkeeper kicks the ball towards his own teammate.
Agent: Goalkeeper.

3. Explain why some of the leaves may get detached from a tree if we vigorously shake its branch.

This happens due to inertia of rest. When we shake the branch, the branch comes into motion, but the leaves tend to remain in their state of rest due to inertia. If the force of shaking is strong enough, this resistance causes the leaves to get detached and fall.

4. Why do you fall in the forward direction when a moving bus brakes to a stop and fall backwards when it accelerates from rest?

Falling forward: When a moving bus brakes, the lower part of your body in contact with the bus comes to rest, but the upper part of your body tends to remain in motion due to the inertia of motion. This causes you to fall forward.

Falling backward: When a bus accelerates from rest, the lower part of your body moves forward with the bus, but the upper part tends to remain at rest due to the inertia of rest. This causes you to fall backwards.

Exercises

1. 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.

Yes, it is possible.
According to Newton's First Law of Motion, if the net external force is zero, an object at rest remains at rest, and an object in motion continues to move with a uniform velocity.
Conditions:
1. The object must be moving with a constant speed.
2. The object must be moving in a straight line (constant direction).
Basically, the object must have a constant velocity.

2. When a carpet is beaten with a stick, dust comes out of it. Explain.

This is due to inertia of rest. When the carpet is beaten, the fibers of the carpet move suddenly, but the dust particles trapped in it tend to remain in their state of rest. As the carpet moves away from the dust particles, they get separated and fall out.

3. Why is it advised to tie any luggage kept on the roof of a bus with a rope?

Luggage is tied to preventing it from falling due to inertia.
1. When the bus starts suddenly, the luggage tends to stay at rest (inertia of rest) and may fall backwards.
2. When the bus stops suddenly, the luggage tends to continue moving (inertia of motion) and may fall forward.
3. When the bus turns sharply, the luggage tends to continue in a straight line (inertia of direction) and may fall sideways.

4. 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 it would want to come to rest.

Answer: (c) there is a force on the ball opposing the motion.
Explanation: The force of friction between the ground and the ball acts in the direction opposite to the motion, causing it to slow down and stop.

5. A truck starts from rest and rolls down a hill with a constant acceleration. It travels a distance of 400 m in 20 s. Find its acceleration. Find the force acting on it if its mass is 7 tonnes (Hint: 1 tonne = 1000 kg).

Given:
Initial velocity (u) = 0 m/s
Distance (s) = 400 m
Time (t) = 20 s
Mass (m) = 7 tonnes = 7 × 1000 = 7000 kg

Step 1: Find Acceleration (a)
Using the second equation of motion: s = ut + ½at²
400 = (0 × 20) + ½ × a × (20)²
400 = 0 + ½ × a × 400
400 = 200a
a = 400 / 200
a = 2 m/s²

Step 2: Find Force (F)
Using Newton's Second Law: F = ma
F = 7000 kg × 2 m/s²
F = 14000 N

6. A stone of 1 kg is thrown with a velocity of 20 m s^-1 across the frozen surface of a lake and comes to rest after travelling a distance of 50 m. What is the force of friction between the stone and the ice?

Given:
Mass (m) = 1 kg
Initial velocity (u) = 20 m/s
Final velocity (v) = 0 m/s (comes to rest)
Distance (s) = 50 m

Step 1: Calculate Acceleration (a)
Using the third equation of motion: v² - u² = 2as
(0)² - (20)² = 2 × a × 50
-400 = 100a
a = -400 / 100
a = -4 m/s² (Retardation)

Step 2: Calculate Force of Friction (F)
F = ma
F = 1 × (-4)
F = -4 N
The negative sign indicates the force opposes the motion. Magnitude is 4 N.

7. 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:
(a) the net accelerating force;
(b) the acceleration of the train; and
(c) the force of wagon 1 on wagon 2.

Given:
Mass of engine = 8000 kg
Mass of 1 wagon = 2000 kg
Number of wagons = 5
Total mass of wagons = 5 × 2000 = 10000 kg
Force of Engine (F_engine) = 40000 N
Friction Force (F_friction) = 5000 N

(a) Net Accelerating Force (F_net)
F_net = F_engine - F_friction
F_net = 40000 - 5000 = 35000 N

(b) Acceleration of the train (a)
Total Mass to be accelerated (Train) = Mass of Engine + Mass of Wagons
Wait, usually "train" implies the whole system, but often in physics problems, we accelerate the whole mass. Let's verify standard interpretation: The engine accelerates the entire mass (itself + wagons).
Total Mass = 8000 + 10000 = 18000 kg.
a = F_net / Total Mass = 35000 / 18000 = 1.944 m/s²

(c) Force of Wagon 1 on Wagon 2
Wagon 1 has to pull Wagons 2, 3, 4, and 5 (4 wagons behind it).
Mass to be pulled = 4 × 2000 = 8000 kg.
Force = Mass × Acceleration
F = 8000 × 1.944 = 15552 N
(Alternative precise calculation using fraction 35/18: 8000 * 35/18 = 15555.5 N)

8. 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?

Given:
Mass (m) = 1500 kg
Acceleration (a) = -1.7 m/s²

Calculation:
Force (F) = ma
F = 1500 × (-1.7)
F = -2550 N

The force must be 2550 N (acting in the direction opposite to motion).

9. What is the momentum of an object of mass m, moving with a velocity v?
(a) (mv)² (b) mv² (c) ½ mv² (d) mv

Answer: (d) mv
Momentum (p) is defined as the product of mass and velocity.

10. 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?

For the cabinet to move at a constant velocity, the net force acting on it must be zero (acceleration = 0).
This means the applied force must be exactly equal and opposite to the frictional force.
Applied Force = 200 N
Therefore, Friction Force = -200 N (magnitude is 200 N, opposite direction).

11. Two objects, each of mass 1.5 kg, are moving in the same straight line but in opposite directions. The velocity of each object is 2.5 m s^-1 before the collision during which they stick together. What will be the velocity of the combined object after collision?

Given:
Mass of object 1 (m₁) = 1.5 kg
Velocity of object 1 (v₁) = 2.5 m/s
Mass of object 2 (m₂) = 1.5 kg
Velocity of object 2 (v₂) = -2.5 m/s (opposite direction)

Conservation of Momentum:
Total momentum before collision = Total momentum after collision
m₁v₁ + m₂v₂ = (m₁ + m₂)V
(1.5 × 2.5) + (1.5 × -2.5) = (1.5 + 1.5)V
3.75 - 3.75 = 3V
0 = 3V
V = 0 m/s

The combined object will come to rest. Velocity = 0 m/s.

12. 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.

The student's logic that the forces "cancel each other" is incorrect because the action and reaction forces act on two different objects (Action: We push truck. Reaction: Truck pushes us). Forces only cancel if they act on the same object.

Why the truck does not move:
The truck does not move because the force applied by the student is not sufficient to overcome the huge force of static friction acting between the truck's tires and the road. The applied force is balanced by the friction force, not the reaction force.

13. 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 magnitude of change of momentum occurred in the motion of the hockey ball by the force applied by the hockey stick.

Given:
Mass (m) = 200 g = 0.2 kg
Initial velocity (u) = 10 m/s
Final velocity (v) = -5 m/s (returns along original path, so opposite sign)

Calculation:
Initial Momentum (p_i) = mu = 0.2 × 10 = 2 kg m/s
Final Momentum (p_f) = mv = 0.2 × (-5) = -1 kg m/s
Change in Momentum = p_f - p_i
Change = -1 - 2 = -3 kg m/s

The magnitude of change of momentum is 3 kg m s^-1.

14. A bullet of mass 10 g travelling horizontally with a velocity of 150 m s^-1 strikes a stationary wooden block and comes to rest in 0.03 s. Calculate the distance of penetration of the bullet into the block. Also calculate the magnitude of the force exerted by the wooden block on the bullet.

Given:
Mass (m) = 10 g = 0.01 kg
Initial velocity (u) = 150 m/s
Final velocity (v) = 0 m/s
Time (t) = 0.03 s

Step 1: Calculate Acceleration (a)
v = u + at
0 = 150 + a(0.03)
-150 = 0.03a
a = -150 / 0.03 = -5000 m/s²

Step 2: Calculate Distance (s)
v² - u² = 2as
0 - (150)² = 2 × (-5000) × s
-22500 = -10000s
s = 22500 / 10000 = 2.25 m

Step 3: Calculate Force (F)
F = ma
F = 0.01 × (-5000) = -50 N
Magnitude of force is 50 N.

15. An object of mass 1 kg travelling in a straight line with a velocity of 10 m s^-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.

Given:
Mass of object (m₁) = 1 kg, Velocity (v₁) = 10 m/s
Mass of block (m₂) = 5 kg, Velocity (v₂) = 0 m/s

(a) Total Momentum Before Impact:
p_total = m₁v₁ + m₂v₂
p_total = (1 × 10) + (5 × 0) = 10 kg m/s

(b) Total Momentum After Impact:
According to the Law of Conservation of Momentum, momentum is conserved.
Momentum After = Momentum Before = 10 kg m/s

(c) Velocity of Combined Object (V):
Total Mass = m₁ + m₂ = 1 + 5 = 6 kg
Momentum = Total Mass × V
10 = 6 × V
V = 10 / 6 = 1.67 m/s

16. An object of mass 100 kg is accelerated uniformly from a velocity of 5 m s^-1 to 8 m s^-1 in 6 s. Calculate the initial and final momentum of the object. Also, find the magnitude of the force exerted on the object.

Given:
Mass (m) = 100 kg
Initial velocity (u) = 5 m/s
Final velocity (v) = 8 m/s
Time (t) = 6 s

(a) Initial Momentum (p_i)
p_i = mu = 100 × 5 = 500 kg m/s

(b) Final Momentum (p_f)
p_f = mv = 100 × 8 = 800 kg m/s

(c) Force (F)
F = Change in Momentum / Time
F = (p_f - p_i) / t
F = (800 - 500) / 6
F = 300 / 6 = 50 N