MCQs with Explanation

3.1 The ratio of angular speed of minute’s hand and hour’s hand of watch is:
(a) 1 : 6
(b) 6 : 1
(c) 1 : 12
✅ (d) 12 : 1
Explanation:
 Hour hand completes 1 round in 12 hours → ωh=2π / 12
 Minute hand completes 1 round in 1 hour → ωm=2π
 Ratio = ωm / ωh=2π / 2π/12=12:1

3.2 A body traveling in a circle at constant speed:
(a) has constant velocity
✅ (b) has an inward radial acceleration
(c) is not accelerated
(d) has an outward radial acceleration
Explanation:
Even though speed is constant, velocity changes due to change in direction.
Hence, there’s an inward centripetal acceleration acting toward the center.

3.3 The tension in the string is minimum when the stone is:
✅ (a) at the top of the circle
(b) half way down
(c) at the bottom of the circle
(d) anywhere in the circle
Explanation:
At the top, gravity and tension both act downward → tension is least.
At the bottom, gravity acts opposite to tension → tension is highest there.

3.4 Every point of a rotating rigid body has:
✅ (a) same angular velocity
(b) same linear velocity
(c) same linear acceleration
(d) same linear distance
Explanation:
All points rotate together, so angular velocity is same.
Linear velocity depends on distance from center → it varies.

3.5 The minimum velocity to put a satellite into orbit is called:
(a) terminal velocity
✅ (b) critical velocity
(c) artificial velocity
(d) angular velocity
Explanation:
Critical velocity is the exact speed needed for a satellite to remain in circular orbit without
falling back to Earth.

3.6 An astronaut in orbit:
(a) will be in a state of weightlessness with respect to capsule
(b) is freely falling towards the Earth

(c) a ball projected at an angle has a straight line path as observed by him
✅ (d) all the above
Explanation:
 The capsule and astronaut are in free fall → weightlessness
 Everything inside moves similarly → appears stationary
 A thrown ball follows a straight line relative to the astronaut

3.7 An object makes 10 revolutions in 2 seconds.
(a) Its period is 2.0 s
(b) Its period is 20 s
✅ (c) Its frequency is 5 Hz
(d) Its frequency is 0.2 Hz
Explanation:
Frequency=No. of revolutions / Time=10 / 2=5 Hz
3.8 A man inside the artificial satellite feels weightlessness because the force of attraction due to the Earth is:
(a) zero at pole
(b) balanced by the force of attraction due to the moon
✅ (c) equal to the centripetal force
(d) non-effective due to some particular design of the satellite
Explanation:
In orbit, the gravitational pull provides the centripetal force, so both astronaut and satellite are
in free-fall → weightlessness.

3.9 When soda bottle is swung in vertical circle, bubbles collect:
(a) Near the bottom
(b) In the middle
(c) Bubbles remain distributed
✅ (d) Near the neck of the bottle

Explanation:
At the top of the circle, apparent gravity is reduced → bubbles rise and gather near the neck
(topmost part).

3.10 Moment of inertia depends on:
(a) mass of the body and its distribution about axis of rotation
(b) volume of the body
(c) kinetic energy
(d) angular momentum
� Explanation:
Moment of Inertia I=∑mr^2 → depends on mass and distance from axis.

 Short Answer Questions

3.1 State second law of motion in case of rotation.
The second law in rotational motion is:
τ=Iα
Where:
 τ\tauτ: torque
 I: moment of inertia
 α\alphaα: angular acceleration

3.2 What is the effect of changing the position of a diver while diving?
When a diver tucks in (reduces body radius), moment of inertia decreases and angular
velocity increases (due to conservation of angular momentum). This helps in completing more
spins before entering water.

3.3 How do we get butter from milk?

By churning milk, we apply circular motion → heavier particles (liquid) move outward, lighter
ones (butter fat) come to center and separate due to centrifugal force.

3.4 Mass is a measure of inertia in linear motion. What is its analogue in rotation?
In rotation, Moment of Inertia (I) is analogous to mass.
It tells how much torque is needed to produce angular acceleration.

3.5 Why is it harder for a car to take turn at higher speed?
Centripetal force F=mv^2r increases with square of speed.
So more friction is required, making it harder (and more dangerous) to turn at high speed.

3.6 Benefits of double rear tires on heavy vehicles:
Double tires:
 Increase contact area → better grip
 Distribute load evenly → safer turning
 Provide stability and reduce risk of overturning

3.7 When car turns left, in which direction do occupants fall?
They tend to fall right (opposite to direction of turn) due to inertia. Their body tries to move in
a straight line while car turns left.

3.8 Why is acceleration of circular motion directed towards center?
This is called centripetal acceleration.
It keeps the object in circular path by pulling it continuously toward center of rotation.

3.9 Why does astronaut feel weightless while orbiting Earth?

Both astronaut and spacecraft are in free-fall around Earth.
No normal force acts on body → sensation of weightlessness.

Constructed Response Question

3.1 If angular velocity of different particles of a rigid body is constant, will the linear velocity of these particles also be constant?
Answer:
No, the linear velocity of different particles will not be constant.
Explanation:
In a rigid body rotating with constant angular velocity ω\omegaω, the linear velocity vvv of a
particle is given by:
v=rω
 r = distance of particle from the axis of rotation
 Since ω is constant but r varies for different particles,
 Linear velocity v will differ for each particle depending on its distance from the axis.

3.2 A loaf of bread is lying on a rotating plate. A crow takes away the loaf of bread and the plate’s rotation increases. Why?
Answer:
This is due to the law of conservation of angular momentum.
Explanation:
 When the crow removes the bread, the moment of inertia I of the system decreases.
 Since there is no external torque, angular momentum L remains conserved:
L=Iω=constant
 Therefore, if I↓, then ω↑ (angular velocity increases).
Result: The plate rotates faster after the bread is removed.
3.3 Why do we tumble when we take the sharp turn with large speed?
Answer:
When taking a sharp turn at high speed, the centripetal force required increases.

If friction between the feet (or tires) and the ground is not enough to provide this force, the body
fails to stay in circular path and tumbles outward due to inertia.

3.4 What will be the time period of a simple pendulum in an artificial satellite?
Answer:
In an artificial satellite, objects are in free-fall (microgravity).
Since g=0g = 0g=0, and:
T=2π √l/g⇒T=∞
So, the pendulum will not oscillate. No time period can be defined.

3.5 Is the motion of a satellite in its orbit, uniform or accelerated?
Answer:
The speed of satellite is constant, but direction continuously changes, so velocity changes.
Hence, motion is uniform in speed but accelerated due to centripetal acceleration.

3.6 What are the advantages that radian has over degree as SI unit?
Answer:
 Radian is a pure ratio (arc length/radius) → no units needed
 Makes equations in circular motion dimensionally simpler
 Used directly in calculus and physics formulas
✅ Therefore, radian is preferred over degree in SI system.

3.7 In uniform circular motion, what are the average velocity and acceleration for one revolution?
Answer:
 Average velocity over full revolution = 0, because displacement = 0
 Average acceleration is also 0 for same reason
Displacement is vector → returns to starting point.

3.8 In a rainstorm with strong wind, where should umbrella be held?
Answer:
Umbrella should be held in the direction opposite to resultant velocity of rain.
If rain falls vertically + wind blows horizontally → hold umbrella tilted forward to block
diagonal direction.

3.9 A ball is just supported by a string without breaking. It breaks during vertical circular motion. Why?
Answer:
During vertical circular motion, especially at bottom of circle, the tension in string becomes:
T=mg+mv^2 / r
This is greater than just mg, so it exceeds limit and breaks the string.

3.10 How is centripetal force supplied in the following cases?
✅ (a) Satellite orbiting Earth:
Gravitational force between Earth and satellite provides centripetal force.
✅ (b) Car turning on level road:
Friction between tires and road acts as centripetal force.
✅ (c) Stone in circular motion using string:
Tension in string acts as centripetal force.

Comprehansive Question

3.1 What is meant by angular momentum? Explain the law of conservation of angular momentum with daily life examples.

Definition of Angular Momentum (L):

Angular momentum is the rotational analogue of linear momentum.
It is the quantity of motion possessed by a rotating body.
L=Iω
Where:
 L = Angular momentum
 I = Moment of inertia
 ω = Angular velocity
 Law of Conservation of Angular Momentum:
“If no external torque acts on a system, the total angular momentum of the system remains
constant.”
L=constant⇒I1ω1=I2ω2
 Derivation / Mathematical Form:
If τext=0, then:
dL / dt=0⇒L=constant
 Daily Life Examples:
1. Figure Skater:
When a skater pulls arms inward, moment of inertia I decreases → angular velocity ω increases
→ skater spins faster.
2. Ice Dancer / Diver in Air:
A diver curls the body to reduce I and spin faster before landing.
3. Neutron Star:
A dying star collapses, reducing radius → moment of inertia decreases → it spins extremely fast.
4. Spinning Plate & Bread (Crow example):
When bread is removed, moment of inertia I↓, so ω↑ → plate spins faster
3.2 Show that orbital angular momentum; L=mvr

Consider a particle of mass mmm moving in a circular orbit of radius rrr with linear speed v.
Angular Momentum:L=r×p=r×mv=mvr(since r⊥v)
Hence proved:
L=mvr
Where:
 L = Angular momentum
 m = Mass
 v = Linear velocity
 r = Radius from axis

✅ 3.3 Define Moment of Inertia. Prove that torque = Iα
Definition:
Moment of inertia III is the measure of resistance of a rotating body to changes in its angular
velocity.
I=∑ m r^2
Torque and Angular Acceleration:
From Newton’s second law for rotation:
τ=Iα
Derivation:
Let a mass mmm rotates in a circle of radius rrr:
τ=r⋅F=r⋅ma=r⋅m(rα)=mr2α⇒τ=Iα

3.4 What are artificial satellites? Calculate minimum time period to orbit.
Definition:
Artificial satellites are man-made objects that orbit Earth or any celestial body. Example: GPS,
weather satellites.
Minimum Time Period (Near Earth):
Use the formula:

T=2π √r^3 / GM
For lowest orbit (just above Earth): r≈R=6.4×10^6  m
Gravitational constant: G=6.67×10^−11
Earth mass: M=5.97×10^24
T=2π √(6.4×10^6)^3 / 6.67×10^−11⋅5.97×10^24 ⇒T≈84.5 minutes

3.5 Define orbital velocity. Derive its expression.
Definition:
The minimum velocity required by a satellite to orbit Earth without falling back due to gravity.
Centripetal force = Gravitational force:
Mv^2 / r=GMm / r^2⇒v= √GM / r=Orbital velocity
For near-Earth orbit r=R=6.4×10^6 m:
v= √6.67×10^−11⋅5.97×10^24 / 6.4×10^6⇒v≈7.9 km/s

3.6 Write a note on artificial gravity. Derive expression for frequency.
Artificial gravity:
Created by rotating a spaceship. The normal reaction provides centripetal force that simulates
gravity.
F=mω^2r=mg⇒ω= √g / r
Frequency (f):
f=ω / 2π=1 / 2π √g / r⇒f=1 / 2π √g / r

✅ 3.7 Prove that: (i) v =rω, (ii) a=rα
(i) Linear and angular velocity relation:
Arc length: s=rθ⇒ds / dt=rdθ / dt=rω⇒v=rω
(ii) Linear and angular acceleration relation:
Dv / dt=rdω / dt=rα⇒a=rα