Vertical Circular Motion: Comprehensive NEET Physics Notes

1. Vertical Circular Motion

1.1 Introduction to Vertical Circular Motion

Vertical circular motion refers to the movement of an object along a circular path under the influence of gravity, where the motion occurs in a vertical plane. This is commonly seen in systems like a pendulum, a roller coaster loop, or a stone tied to a string and swung in a vertical circle. Understanding this motion is important for solving NEET problems related to energy conservation, variations in tension, and changes in velocity at different points in the circular path.

In this motion, the object experiences two key forces:

Gravitational Force ( $m g$ ): Acts downward throughout the motion.
Tension (or Normal Force) ( $T$ ): Varies at different points in the path, with its magnitude dependent on the object's position.

1.2 Forces in Vertical Circular Motion

The forces acting on a body in vertical circular motion vary depending on its position in the loop. The two critical positions to consider are the highest point (Point B) and the lowest point (Point A).

Case 1: At the Lowest Point (Point A)

At the lowest point, the tension in the string (or normal force) is at its maximum. The forces acting on the object are:

Tension ( $T_{A}$ ), acting upward.
Weight ( $m g$ ), acting downward.

According to Newton's second law of motion:

$T_{A} - m g = \frac{m v _{A}^{2}}{r}$

where:

$v_{A}$ is the speed at the lowest point,
$r$ is the radius of the circular path,
$m$ is the mass of the object.

Thus, the tension at the lowest point becomes:

$T_{A} = m g + \frac{m v _{A}^{2}}{r}$

Case 2: At the Highest Point (Point B)

At the highest point, both the tension and the weight act downward. The forces are described by:

$T_{B} + m g = \frac{m v _{B}^{2}}{r}$

At this point, the tension is at its minimum. If the tension becomes zero, the object will just make the loop, which gives us the minimum speed at the top:

$v_{B} = g r$

1.3 Energy Conservation in Vertical Circular Motion

In vertical circular motion, energy is conserved if we ignore air resistance and friction. The total mechanical energy (sum of kinetic energy and potential energy) remains constant throughout the motion.

At any point in the motion:

$K + U = constant$

where:

Kinetic Energy (K) = $\frac{1}{2} m v^{2}$
Potential Energy (U) = $m g h$

Case 1: At the Lowest Point (Point A)

The kinetic energy is maximum at the lowest point, while the potential energy is zero (taking this as the reference level):

$E_{A} = \frac{1}{2} m v_{A}^{2}$

Case 2: At the Highest Point (Point B)

At the highest point, the kinetic energy is at its minimum, and the potential energy is maximum:

$E_{B} = \frac{1}{2} m v_{B}^{2} + 2 m g r$

Using energy conservation between points A and B:

$E_{A} = E_{B}$

1.4 Critical Speed for Vertical Circular Motion

The critical speed is the minimum speed the object must have at the lowest point to complete the loop. Using the principle of energy conservation:

$\frac{1}{2} m v_{A}^{2} = \frac{1}{2} m v_{B}^{2} + 2 m g r$

Substitute the minimum speed at the top ( $v_{B} = g r$ ) into the equation:

$\frac{1}{2} m v_{A}^{2} = \frac{1}{2} m (g r) + 2 m g r$

Solving this yields the critical speed at the bottom of the loop:

$v_{A} = 5 g r$

1.5 Tension Variation in Vertical Circular Motion

The tension in the string (or normal force) varies throughout the motion:

At the lowest point, tension is maximum: $T_{A} = m g + \frac{m v _{A}^{2}}{r}$ .
At the highest point, tension is minimum and can be zero when $T_{B} = 0$ , meaning the object is just making the loop.
In between these points, tension can be calculated using the forces and speeds specific to each position.

Real-Life Application:

A real-life example of vertical circular motion is the loop-the-loop in a roller coaster. The speed of the coaster at the lowest and highest points follows the same principles, ensuring it has enough velocity to complete the loop safely.

NEET Tip:

For NEET problems, always focus on energy conservation to find the speed at various points. If tension or normal force is asked, apply Newton’s second law considering the forces acting at the specific point in the circular path.

Common Misconception:

Students often assume that the speed at the top of the loop can be zero. In fact, the object must have a minimum speed ( $v_{B} = g r$ ) at the top to stay in motion, or else it will not complete the loop.

Quick Recap:

Vertical circular motion involves both gravitational force and tension (or normal force).
The minimum speed at the highest point is $v_{B} = g r$ .
The critical speed at the lowest point is $v_{A} = 5 g r$ .
Use energy conservation to solve for speed and tension at different points in the loop.

Practice Questions:

A 0.5 kg stone is tied to a string and rotated in a vertical circle of radius 1.5 m. What is the minimum speed the stone must have at the bottom to complete the loop?
Solution: The critical speed at the bottom is given by: $v_{A}$