Circular motion MCQ Quiz - Objective Question with Answer for Circular motion - Download Free PDF

Solution:

Let the radius of the disc be r₀ and the time period of rotation be T = 2π / ω.

Let the pebble Q be at the point (0, y₀) at the time of projection. For projectile motion, let r be the range and t be the time of flight. It is given that t < T / 8 and r < r₀ / 2.

The rotation of the disc provides additional initial velocity to the pebbles given by:

ΔūQ = −y₀ω î and ΔūP = r₀ω î

These additional velocities are perpendicular to the plane of projection. The time of flight t is not affected by these velocities. However, they introduce additional displacements in the x-direction:

Δr̄Q = −y₀ωt î and Δr̄P = r₀ωt î

So the new landing points Q₁ and P₁ shift by these respective distances.

From triangle OQ₁Q′₁:

θ = tan⁻¹ (y₀ωt / (y₀ + r)) ≤ tan⁻¹ (ωt) ≤ ωt

So the angle θ is less than the angle through which the disc rotates in time t. Hence, landing point Q′₁ lies in the unshaded region.

For pebble P, the horizontal displacement is r₀ωt, and the subtended angle is also ωt. But from the geometry of the disc, the angle of point P′₁ is greater than ωt. Therefore, P′₁ also lies in the unshaded region.

Answer: (C) Both P and Q land in the unshaded region.

CONCEPT:

Time Period in Circular Motion

• The time period (T) of a circular motion is the time taken to complete one full revolution.
• It is related to the angular velocity (ω) by the formula:

  T = ^2π/_ω

EXPLANATION:

• Angular velocity (ω) = 2π rad/s
• Using the formula:
  • T = ^2π/_ω
  • T = ^2π/_2π
  • T = 1 second

Therefore, the time period (T) of the car's circular motion is 1 second.

CONCEPT:

• Centripetal acceleration is defined as the ratio of the square of the velocity and radius and it is written as,

           ac = \(\frac{v^2}{R}\) 

           here we have a_c as the centripetal acceleration, v is the velocity and R is the radius.

CALCULATION:

A ball is released from rest from point P of a smooth semi-spherical vessel as shown in the figure below,

Using the trigonometry identity to find h we have;

\(sin\alpha = \frac{h}{R}\)

⇒ h = R sin\(\alpha\)

and the first law of motion we have;

v² - u² = 2gh

Here is the initial velocity, u = 0 m/s we have;

v2 = 2gh

⇒ v = \(\sqrt{2gh}\) 

⇒ v = \(\sqrt{2gRsin\alpha}\) -----(1)

The centripetal acceleration is written as;

a_c = \(\frac{v^2}{R}\) 

Now, on putting the value of equation (1) above we have,

ac = \(\frac{2gRsin\alpha}{R}\) 

⇒ a_c = 2g sin\(\alpha\) -----(2)

At point Q we can see that normal N and force F which mg sin\(\alpha\) we have;

N -  mg sin α  = ma_c

⇒ N = ma_c + mg sinα ----(3)

Now, on putting the value of equation (2) in equation (3) we have;

N = m × 2 g sin α + mg sin α 

N = 3 × mg sin α 

The ratio of the centripetal force and normal reaction is written as;

\(\frac{ma_c}{N} = \frac{ 2mgsin\alpha}{3mg sin\alpha}\)

⇒ \(\frac{ma_c}{N} = \frac{2}{3}\) = constant

Therefore, the ratio of the centripetal force and normal reaction is constant.

Hence, option (3) is the correct answer.

Calculation

From the above diagram, we have

N = mg cosθ + (mv² / R) sinθ                        (1)

f_max = μN ⇒ f_max = μ_s mg cosθ + (μ_s mv² / R) sinθ

mg sinθ + f_max = (mv² / R) cosθ                     (2)

Putting the value

mg sinθ + μ_s mg cosθ + (μ_s mv² / R) sinθ = (mv² / R) cosθ

g sinθ + μ_s g cosθ = (v² / R)(cosθ − μ_s sinθ)

gR [ (tanθ + μ_s) / (1 − μ_s tanθ) ] = v²

v = √[ gR (tanθ + μ_s) / (1 − μ_s tanθ) ]

Calculation:

Work energy theorem 

\(\mathrm{Mg}(\mathrm{R}+\mathrm{R} \cos 60)+\frac{1}{2} \mathrm{k}\left(\mathrm{R}^{2}-0^{2}\right)=\frac{1}{2} \mathrm{mv}^{2}\)

\(\mathrm{Mg} \frac{3 \mathrm{R}}{2}+\frac{\mathrm{KR}^{2}}{2}=\frac{1}{2} \mathrm{mv}^{2}\)

\(V=\sqrt{3 g R+\frac{K R^{2}}{m}}\)

Explanation:

Centripetal Force: It is a force required to move a body uniformly in a circular motion. This force acts along the radius and towards the center of the circle.

• When a body moves in a circle, its direction of motion at any instant is along the tangent of a circle. But according to Newton’s first law of motion, A body cannot change its direction of itself an external force is required for this purpose. This external force is the centripetal force

\({\bf{Centripetal}}\;{\bf{Force}}\;\left( {\bf{F}} \right) = \frac{{m{v^2}}}{r}\;\left[ {{\rm{m}} = {\rm{mass}},{\rm{\;v}} = {\rm{velocity}},{\rm{\;r}} = {\rm{radius}}} \right]\)

• The centripetal force required for circular motion along the surface of the road, towards the center of the turn. The Static friction between tire and road provides the necessary centripetal force.

Explanation:

Distance: 

• It is defined as the length of the path traveled by a body. 
• It is a scalar quantity.
• It cannot be in negative value.
• It is non-zero in circular motion

​Displacement

• It is the shortest distance between the initial and final position of the particle.
• It is a vector quantity.
• It can be positive, negative and zero.
• It is zero in circular motion.

When an object moves in a straight line without changing the direction the distance and displacement will be equal in magnitude.

• When an object changes its direction during the motion its path length will become more compared to the distance between the initial and the final position, so in this case, the magnitude of distance becomes more than the displacement.
• So the distance will always be more than or equal to displacement.

The correct answer is option 4) i.e. all of the above

CONCEPT:

• Circular motion: The motion of an object along a circular path is called as circular motion.
• Centripetal force: It is a net force acting on an object undergoing a circular path of motion, such that the force acts in a direction towards the centre of curvature.

​EXPLANATION:

• A circle is assumed to be a polygon with infinitely many sides such that each side approximates to a point.
• So, an object moving on a circular path undergoes a change in direction at every point. 
• Since direction changes at every point, velocity changes at every point.
• Also, a centripetal force always acts on an object under circular motion since it is the force that makes a body follow a curved path.

The correct answer is option 4) i.e. All of the above

CONCEPT:

• Uniform circular motion is where a moving object traces a circular path with constant speed.
  • ​A circle is assumed to be a polygon with infinitely many sides such that each side approximates to a point.
  • So, the object moving on a circular path undergoes a change in direction at every point.
  • Since direction changes and speed remains constant, velocity is varying.
• Kinetic energy is the measure of energy possessed by a moving object.

It is given by the equation KE = \(\frac{1}{2}mv^2\)

EXPLANATION:

Given object is under uniform circular motion.

• An object under uniform circular motion has a constant speed.
• In a uniform circular motion, the magnitude of velocity is uniform but the direction keeps changing at every point. So, the magnitude of acceleration is constant.
• Kinetic energy is proportional to the magnitude of the velocity of a moving object ⇒ KE ∝ v². Hence, kinetic energy remains constant in a uniform circular motion.

Confusion Points

• Only the magnitude of velocity is considered for kinetic energy. 
• Kinetic energy is a scalar quantity. 

The correct answer is option 4) i.e. both acceleration and velocity changes.

CONCEPT:

• Uniform motion is the type of motion where a moving object traces equal distances in equal intervals of time.
  • ​Since the distance and time intervals are the same, speed is constant in uniform motion.
• Uniform circular motion is where a moving object traces a circular path with constant speed.
  • ​A circle is assumed to be a polygon with infinitely many sides such that each side approximates to a point.
  • So, if the object moving on a circular path undergoes a change in direction at every point.
  • Since direction changes and speed remains constant, velocity is varying.

EXPLANATION:

• Velocity is changing in a uniform circular motion as the direction of the object keeps changing at every point.
• Acceleration is the rate of change of velocity. Since velocity keeps changing at every instant, acceleration also changes.

Therefore, both acceleration and velocity changes in a uniform circular motion.

Concept:

Circular Motion: The movement of an object along a circumference of a circle or rotation along a circular path is called circular motion.

• Uniform circular motion: The circular motion in which the speed of the particle remains constant is called uniform circular motion. In a uniform circular motion, force supplies the centripetal acceleration.
  • The kinetic energy and the speed of the body remain constant.

​Centripetal Force: 

It is a force required to move a body uniformly in a circle. This force acts along the radius and towards the centre of the circle.

\({\bf{Centripetal}}\;{\bf{Force}}\;\left( {\bf{F}} \right) = \frac{{m{v^2}}}{r}\)

• The centripetal force required for circular motion along the surface of the road, towards the centre of the turn. 

Explanation:

• The Centripetal force is required to maintain a body in Uniform Circular Motion. So this is correct.
• Centripetal Force acts when a body moves around the other body, the force that causes this acceleration and keeps the body moving along a circular path is acting towards the center

CONCEPT:

• Circular Motion: Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path.
  •  The force acts continuously at right angles to the velocity of the particle.
• Uniform circular motion: The circular motion in which the speed of the particle remains constant is called uniform circular motion. In a uniform circular motion, force supplies the centripetal acceleration.

• In a uniform circular motion, a body is experiencing a force in the inward direction.
• For example: the motion of the hands of a clock, motion of the moon revolving around the earth.

EXPLANATION:

• From the above, it is clear that the movement of the second hand of a watch is an example of uniform circular motion. Therefore option 1 is correct.
• The motion of a giant wheel is a rotatory motion because all the objects within the wheel move in a circular manner in the same direction around a fixed axis. Therefore option 4 is incorrect.

CONCEPT:

• Circular Motion:  Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path.
•  The force acts continuously at right angles to the velocity of the particle.
• Uniform circular motion: The circular motion in which the speed of the particle remains constant is called uniform circular motion.
• In a uniform circular motion, force supplies the centripetal acceleration.

ac = v2/r, where ac is centripetal acceleration, v is velocity, r is the radius.

• The speed and kinetic energy of the particle remains constant.

EXPLANATION:

• If the Force F acts on a body and it gets displaced by a distance of S, the work done in that case is given by:

⇒ W = FS Cos θ

• In the case of uniform circular motion, Cos 90^o = 0 and that is why W = 0 (The force and displacement are perpendicular to each other).  So option 3 is correct.

Correct option-2

Concept:

Circular Motion

• An object is said to perform a circular motion when it moves on a circular path. 
• While describing circular motion we often use the term circular motion of point object or particle.
• This is because we learn circular motion with the brach of the translation motion of a body on a circular path and disregard any rotation.
• Therefore, we represent the body as a particle.

Types of Circular Motion
Circular motion is classified into two categories
1.Uniform circular motion:

• If a particle moves with a constant speed in a circle, then the motion is called uniform circular motion. 
• In uniform-circular motion, particle possesses only centripetal acceleration (ar).

2.Non-uniform circular motion:

• If the particle moves with variable speed then the motion is called non-uniform motion.
• In uniform-circular motion, particle possesses two kinds of acceleration.

• (i) Centripetal acceleration (ar) &
• (ii) Tangential acceleration (at)

Calculation:

Given:-

\(\frac{r_{1}}{r_{2}}=\frac{4}{9}\) &

\(a_{r_{1}}=a_{r_{2}}\)

In a circular motion, the centripetal acceleration ar is given by:-

\(a_{r}=\frac{v^{2}}{r}\)

where v is linear speed and r is the radius.

As \(a_{r_{1}}=a_{r_{2}}\)

\(\Rightarrow \frac{v_{1}^{2}}{r_{1}}=\frac{v_{2}^{2}}{r_{2}}\)

\(\Rightarrow \frac{v_{1}}{v_{2}}=\sqrt{\frac{r_{1}}{r_{2}}}\)

\(\Rightarrow \frac{v_{1}}{v_{2}}=\sqrt{\frac{4}{9}}=\frac{2}{3}\)

Hence, option-2 is correct

The correct answer is option 4) i.e. kinetic energy.

CONCEPT:

• Uniform circular motion is where a moving object traces a circular path with constant speed.
  • ​A circle is assumed to be a polygon with infinitely many sides such that each side approximates to a point.
  • So, the object moving on a circular path undergoes a change in direction at every point.
  • Since direction changes and speed remains constant, velocity is varying.
• Kinetic energy is the measure of energy possessed by a moving object.

It is given by the equation KE = \(\frac{1}{2}mv^2\)

EXPLANATION:

Given that the object is moving in a circular path with uniform motion. Thus, the object is under uniform circular motion.

• Velocity is changing in a uniform circular motion as the direction of the object keeps changing at every point.
• Acceleration is the rate of change of velocity. Since velocity keeps changing at every instant, acceleration also changes.
• In a uniform circular motion, the magnitude of velocity is uniform but the direction keeps changing at every point. Kinetic energy is proportional to the magnitude of the velocity of a moving object.

Hence, kinetic energy remains unchanged in a uniform circular motion.