Unit 3: Circular and Rotational Motion — Short Questions

The torque acting on a body is equal to the rate of change of its angular momentum.
(OR) The torque acting on a body is equal to the product of moment of inertia and angular acceleration.
Mathematically: τ = Iα

The diver changes his body position to change his moment of inertia I, when he takes off from the diving board, he extends his arms and legs in order to have larger moment of inertia. This will decrease his angular velocity ω. When he pulls his legs and arms close to one another his moment of inertia is reduced. This will result in increase of his angular velocity ω, so as to conserve angular momentum.

Since milk is a mixture of light and heavy particles, when it is rotated, the light particles gather near the -axis of rotation whereas the heavy particles will go outwards and hence, cream can easily be separated from milk.

Moment of inertia is analogue in rotational motion it plays the same role as mass in linear motion.
Newton's second law of motion can be expressed in angular motion as τ = Iα

The centripetal force for a turn is given by Fc = mv²/r
Where m is mass, v is the velocity and r is the radius of the turn.
As velocity (v) increases the required centripetal force increases. This makes it harder to take a turn at high speeds, as more force is needed to keep the car on the curved path.
If the required frictional force is not available, the car may skid.
Hence, turning is harder at higher speed due to increased centripetal force demand.

Heavy vehicles consists of double tyres to increase the surface area in contact with road, distributing the weight more evenly and reducing pressure on individual tyre, this enhances stability, safety and load carrying capacity of vehicles.

When a moving car turns around a corner to the left, the occupants tend to fall to the right due to inertia, actually the bodies of occupants want to continue moving in the original straight line path, causing them to be thrown in opposite direction of turn.

The direction of velocity of the body moving in circular path changes continuously. As velocity is a vector quantity, a change in its direction implies an acceleration. This acceleration is called centripetal acceleration 'and it is always directed towards the centre of circle in the direction of centripetal force and is given by an expression
a_c = v²/r
Which is necessary to keep the body in circular path.

An astronaut feels weightlessness while orbiting, because both the astronaut and spaceship are in the state of free fall towards Earth under the influence of gravity. There is no normal reaction force acting on astronaut.
Therefore, the bodies as well as the astronauts in a satellite find themselves in a state of apparent weightlessness.

The product of mass and square of the distance of the body from the axis of rotation is known as moment of inertia.
Mathematically the moment of inertia is I = mr². Its SI unit is kgm² and its dimension is [ML²].

It is the angle at the centre of circle whose arc distance is equal to radius of the circle.

The work done by the centripetal force is zero because centripetal force is along the radius and displacement is along the tangent to the circular path.
Take W = F⃗.d⃗ = Fd cos 90°
Since cos90° = 0
∴ W = Fd (0) = 0

The outer edge of a road is higher than inner edge is called banking of a road which is done to provide centripetal force to an object during turn.

When a body is moving in a circle of radius r with linear velocity V and angular velocity ω then they are related by the following relation:
v = rω
v is tangential velocity whereas ω is the angular velocity.

When a body is moving in a circle then angle traced by its radial line at the centre of the circle is called angular displacement of the body. It is an experimental fact that for its small values the angular displacement is a vector quantity, and for its large values the angular displacement is a scalar quantity.
When it is a vector then its direction is found by using right hand rule. Its S.I unit is radian.

The force applied on a body to produce torque is:
F = ma
For circular motion a = rα
Therefore F = mrα
Multiply both sides by r
rF = mr²α
by definitions for the case of circular motion we have rF = τ and mr² = I.
Therefore τ = Iα
This is the value of torque in terms of moment of inertia.

The velocity of satellite orbiting very close to earth. Earth i.e it height is very small as compared to the earth radius is known as critical velocity. It can be evaluated by:
V = √(Rg)
= √(6.4×10⁶ × 9.8)
= 7.9×10³ ms^-1
= 7.9 kms^-1

No, the linear velocity of the particles will not be constant because the relation between linear and angular velocity is expressed as v = rω
This shows that the linear velocity (v) depends on the distance (r) from the axis of rotation.
Particles at different distances (r) from the axis will have different linear velocities, even with the same angular velocity (ω).

When the crow takes the loaf of bread, the mass on the plate decreases.
According to the conservation of angular momentum (L = Iω), if the moment of inertia (I = mr²) decreases (due to reduced mass), the angular velocity (ω) increases to conserve angular momentum.
Hence, plate rotation increases when loaf is removed.

When taking a sharp turn at high speed, we tumble due to inertia and need for centripetal force during a sharp turn.
1. Inertia: When we move in a straight line, our body wants to continue in straight line due to inertia (according to Newton's First Law)
2. Centripetal force: When we take sharp turn we require centripetal force (Fc = mv²/r) to change the direction of our motion, we may tumble outward and balance at large speed if the force is insufficient.
To avoid tumbling, we need to slow down or adjust our body position to counteract the centrifugal force.

The time period (T) of a simple pendulum is given by:
T = 2π√(l/g)
Since g ≈ 0 (due to state of weightlessness) the time period (T) would approach infinity.
In other words, the pendulum would not oscillate in the same way it does on Earth's surface.

The motion of a satellite in its orbit is accelerated.
Reason: the Speed of satellite may be constant but its direction is continuously changing due to the gravitational force acting toward the center of the Earth. This change in direction means the velocity is changing, resulting in centripetal acceleration (ac = v²/r).
Hence, motion is called uniform circular motion which is also type of accelerated motion (due to change in direction of velocity).

The main advantages of radians over degrees are:
1. Simpler formulas: Many math and physics formulas are simpler in radians (e.g., arc length S = rθ)
2. Unitless: Radians are dimensionless (a ratio of lengths), making them easier to use in equations.

The average velocity is defined as
Average velocity = Total displacement / Total time
As in one complete revolution
Total displacement = 0 (because initial and final position are same)
So,
Average velocity = 0 / Δt = 0
The average acceleration is defined mathematically by a relation
Average acceleration = Change in velocity / Total time taken
∴ Total change in velocity = 0 (speed is constant during uniform circular motion)
Hence, average acceleration = 0 / Δt = 0.
Thus, average velocity and average acceleration for one revolution during uniform circular motion are zero.

Tilt the umbrella against the direction of the wind, so it blocks the rain blown at an angle to deflect rain and wind effectively.

When the ball is just supported by a string (hanging at rest), the tension needs only to balance its weight.
T = mg
But when ball is whirled is a vertical circle, especially at the bottom of circle, the string must provide both the weight of the ball and centripetal force to keep it moving in circular motion.
Mathematically:
T = mg + mv²/r
The string can hold the ball at rest, but breaks due to the additional centripetal force requirement especially at high speed.

i. The gravitational force provides necessary centripetal force to satellite orbiting around the Earth according to Newton's law of gravitation
F = GMms/r²

ii. When a vehicle takes turn on a road, it also needs centripetal force which is provided by the friction between the tyres and the road.

iii. When a stone is whirled in a horizontal circle with the help of a string, then tension in the string provides necessary centripetal force.
Mathematically:
T = mv²/r