All questions of Circular Motion for EmSAT Achieve Exam

Centrifugal force, a fictitious force, peculiar to a particle moving on a circular path, that has the same magnitude and dimensions as the force that keeps the particle on its circular path (the centripetal force) but points in the opposite direction.

Given, mass of stone, m = 5 kg
Length of string, l = 10 m
Maximum tension, T = 160 N
We need to find the maximum velocity of revolution that can be given to the stone without breaking the string.

Maximum velocity of revolution:
We know that the tension in the string, T is given by:
T = (mv²) / r
where m is the mass of the stone, v is the velocity of the stone and r is the radius of the circle.

Finding the radius:
The length of the string, l is equal to the circumference of the circle, so:
l = 2πr
=> r = l / (2π)
=> r = 10 / (2π) = 1.59 m

Finding the maximum velocity:
Substituting the values in the tension equation, we get:
T = (mv²) / r
=> v² = (Tr) / m
=> v = √[(Tr) / m]

Now, substituting the given values, we get:
v = √[(160 × 1.59) / 5]
=> v = 17.88 m/s

Therefore, the maximum velocity of revolution that can be given to the stone without breaking the string is 17.88 m/s.

A banked turn (or banking turn) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. ... The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to the horizontal.

Circular motion involves the motion of an object in a circular path. The object moves around the center of the circle with a constant speed. In circular motion, the object experiences various types of acceleration and velocity. Let’s discuss the given options to understand circular motion in detail.

Radial acceleration is non-zero:
Radial acceleration is the acceleration of an object towards the center of the circle. In circular motion, the direction of velocity changes continuously, and hence the object experiences a change in direction. This change in direction leads to the acceleration of the object towards the center of the circle, known as radial acceleration. Therefore, option ‘a’ is correct.

Radial velocity is zero:
Radial velocity is the velocity of an object in the radial direction, i.e., towards the center of the circle. In circular motion, the object moves in a circular path without any radial motion, i.e., the object moves perpendicular to the radius of the circle. Hence, radial velocity is zero in circular motion. Therefore, option ‘b’ is incorrect.

Body is in equilibrium:
Equilibrium is a state where the object is at rest or moves with a constant velocity. In circular motion, the object moves with a constant speed, but the direction of the velocity changes continuously. Therefore, the object is not at rest, but it is also not moving with a constant velocity. Hence, the body is not in equilibrium in circular motion. Therefore, option ‘c’ is incorrect.

All of the above:
From the above discussion, we can conclude that radial acceleration is non-zero, radial velocity is zero, and the body is not in equilibrium in circular motion. Therefore, option ‘d’ is the correct answer.

Conclusion:
Circular motion involves the motion of an object in a circular path. The object experiences various types of acceleration and velocity in circular motion. In circular motion, radial acceleration is non-zero, radial velocity is zero, and the body is not in equilibrium.

Understanding Circular Motion
Circular motion can be categorized into two types based on the speed of the object moving along the circular path: uniform and non-uniform circular motion.
Uniform Circular Motion
- Definition: This occurs when an object moves in a circular path at a constant speed.
- Key Characteristics:
- The magnitude of the velocity remains constant.
- The direction of the velocity changes continuously, which implies that the object is accelerating towards the center of the circle (centripetal acceleration).
- An example is a satellite orbiting Earth at a fixed speed.
Non-Uniform Circular Motion
- Definition: This involves an object moving in a circular path but with varying speed.
- Key Characteristics:
- The speed of the object changes; it can either accelerate or decelerate.
- The object experiences both centripetal acceleration (due to changing direction) and tangential acceleration (due to changing speed).
- An example includes a car taking a curve while speeding up or slowing down.
Correct Answer Explanation
The question allows for two correct options:
- Option B: Uniform, Constant
- This describes the scenario where an object maintains a constant speed in a circular path.
- Option C: Non-uniform, Varying
- This describes the scenario where the speed of the object changes while still moving in a circular path.
Both options reflect different characteristics of circular motion, making them valid answers depending on the context. Understanding these types is crucial for analyzing motion in physics.

Understanding Uniform Circular Motion
Uniform circular motion refers to motion along a circular path at a constant speed. In this scenario, the object’s speed remains constant, but its direction changes continuously, resulting in acceleration towards the center of the circle.
Analyzing the Options
- Motion of a racing car on a circular track
- This motion is often not uniform because a racing car changes its speed while navigating turns. Drivers accelerate or decelerate, which means the speed is not constant.
- Motion of the moon around the earth
- The moon moves in a nearly circular orbit at a constant speed, making this a classic case of uniform circular motion.
- Motion of a toy train on a circular track
- If the toy train moves at a constant speed around the track, it exemplifies uniform circular motion.
- Motion of the seconds hand on the circular dial of a watch
- The seconds hand moves at a constant speed, completing a full revolution in a fixed time, thus representing uniform circular motion.
Conclusion
The correct answer is option 'A' – the motion of a racing car on a circular track. This is because the car typically experiences variations in speed, unlike the other options, which maintain a constant speed while moving in a circular path.

Explanation:
Non-uniform circular motion is the motion of an object moving in a circular path with a varying speed. The direction of the velocity and acceleration of an object in non-uniform circular motion changes at every point on the circular path.

Velocity:
Velocity is the rate of change of displacement with respect to time. In non-uniform circular motion, the velocity of the object is always tangent to the circular path. The magnitude of the velocity changes at every point on the circular path.

Acceleration:
Acceleration is the rate of change of velocity with respect to time. In non-uniform circular motion, the acceleration of the object is not constant. It changes at every point on the circular path.

Now, let's discuss the given options one by one:

a) Velocity is radial and acceleration is transverse only:
In this option, the velocity is radial which means it is directed towards the center of the circular path. But the acceleration is transverse which means it is perpendicular to the velocity vector. This option is correct because the direction of the acceleration is always towards the center of the circular path in non-uniform circular motion.

b) Velocity is transverse and acceleration is radial only:
In this option, the velocity is transverse which means it is perpendicular to the radius vector. But the acceleration is radial which means it is directed towards the center of the circular path. This option is incorrect because the direction of the acceleration is not always radial in non-uniform circular motion.

c) Velocity is radial and acceleration has both radial and transverse components:
In this option, the velocity is radial which means it is directed towards the center of the circular path. But the acceleration has both radial and transverse components. This option is incorrect because the direction of the acceleration is always towards the center of the circular path in non-uniform circular motion.

d) Velocity is transverse and acceleration has both radial and transverse components:
In this option, the velocity is transverse which means it is perpendicular to the radius vector. But the acceleration has both radial and transverse components. This option is incorrect because the direction of the acceleration is not always radial in non-uniform circular motion.

Hence, the correct option is 'a' - Velocity is radial and acceleration is transverse only.

**Uniform Circular Motion**

Uniform circular motion refers to the motion of an object traveling in a circular path at a constant speed. In this type of motion, the object moves along the circumference of the circle, maintaining a fixed distance from the center.

**Acceleration in Uniform Circular Motion**

While the speed of the object remains constant in uniform circular motion, it is important to note that the object is still experiencing acceleration. This may seem counterintuitive since we typically associate acceleration with a change in speed. However, in uniform circular motion, the acceleration is directed towards the center of the circle and is known as centripetal acceleration.

**Direction of Motion Changes**

The correct answer to why uniform circular motion is called continuously accelerated motion is that the direction of motion changes. In uniform circular motion, an object continuously changes its direction as it moves along the circular path. This change in direction implies that the object is experiencing acceleration.

**Velocity Remains the Same**

While the object is undergoing acceleration in uniform circular motion, its velocity remains constant. Velocity is a vector quantity that includes both magnitude (speed) and direction. In uniform circular motion, the speed of the object remains constant, but its direction changes. Therefore, the velocity of the object remains the same in terms of magnitude, but the direction of the velocity vector continually changes.

**Centripetal Acceleration**

The centripetal acceleration is the acceleration experienced by an object undergoing uniform circular motion. It is directed towards the center of the circle and is responsible for continuously changing the direction of the object's motion. The magnitude of the centripetal acceleration can be calculated using the formula:

a = v^2 / r

where a is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circular path.

**Conclusion**

In conclusion, uniform circular motion is called continuously accelerated motion because the direction of motion of the object undergoing circular motion continually changes. Although the speed (magnitude of velocity) remains constant, the object experiences centripetal acceleration directed towards the center of the circle. This acceleration is responsible for continuously changing the direction of the object's motion, making it an example of accelerated motion.

Perpendicular Force on a Particle in a Plane

When a particle is acted upon by a force of constant magnitude that is always perpendicular to its velocity, the following can be observed:

Circular Motion

The particle moves in a circular path because the force acts as a centripetal force, pulling the particle towards the center of the circle.

Constant Kinetic Energy

Since the force is always perpendicular to the velocity of the particle, it does not do any work on the particle. Therefore, the kinetic energy of the particle remains constant.

Variable Speed

Although the particle moves in a circular path, its speed is not constant. This is because the force only changes the direction of the particle, not its speed. Therefore, the particle will move faster when it is farther away from the center of the circle and slower when it is closer to the center.

Conclusion

In conclusion, when a particle is acted upon by a force of constant magnitude that is always perpendicular to its velocity, it will move in a circular path with variable speed while maintaining constant kinetic energy.