Vertical Plane Circular Motion

Introduction

Vertical circular motion involves objects moving in a circular path within a vertical plane. This means gravity plays a significant role, and the speed of the object is not constant. We will analyze the forces acting on the object at the highest and lowest points of the circular path using Newton’s Second Law.

KEY TAKEAWAY: Vertical circular motion is non-uniform due to the influence of gravity, making the object’s speed variable.

Forces in Vertical Circular Motion

General Principles

• Newton’s Second Law: The net force acting on an object is equal to the mass of the object multiplied by its acceleration ($F_{net} = ma$). In circular motion, this becomes $F_{net} = \frac{mv^2}{r}$, where $F_{net}$ is the net centripetal force, $m$ is the mass, $v$ is the speed, and $r$ is the radius of the circular path.
• Centripetal Force: The net force always points towards the center of the circle. The forces contributing to the net force can include:
  • Tension (T): Force exerted by a string or rope.
  • Weight (mg): Force due to gravity.
  • Normal Force (N): Force exerted by a surface perpendicular to the object.

Highest Point

At the highest point of the circle, both the weight ($mg$) and tension ($T$) (if a string is involved) or normal force ($N$) (if on a track) point downwards, towards the center of the circle.

• String/Rope:
  $$F_{net} = T + mg = \frac{mv^2}{r}$$
  Where T is the tension in the string, mg is the weight of the object, m is the mass, v is the velocity and r is the radius.
• Track:
  $$F_{net} = N + mg = \frac{mv^2}{r}$$

Lowest Point

At the lowest point of the circle, the weight ($mg$) points downwards, while the tension ($T$) or normal force ($N$) points upwards, towards the center of the circle.

• String/Rope:
  $$F_{net} = T - mg = \frac{mv^2}{r}$$
  Where T is the tension in the string, mg is the weight of the object, m is the mass, v is the velocity and r is the radius.
• Track:
  $$F_{net} = N - mg = \frac{mv^2}{r}$$

Minimum Speed at the Highest Point (for String/Rope)

For an object on a string to maintain circular motion at the highest point, there’s a minimum speed required. If the speed is too low, the string will slacken, and the object will not complete the circle. This occurs when the tension $T = 0$.

$$mg = \frac{mv_{min}^2}{r}$$
$$v_{min} = \sqrt{gr}$$

Velocity when Normal Force is Zero (Track)

When on a track, the formula is the same.
$$v_{min} = \sqrt{gr}$$

EXAM TIP: Always draw a free-body diagram to visualize the forces acting on the object. This will help you correctly apply Newton’s Second Law.

Problem-Solving Steps

1. Draw a Free-Body Diagram: Represent the forces acting on the object at the point of interest (highest or lowest).
2. Choose a Coordinate System: Usually, the direction towards the center of the circle is chosen as positive.
3. Apply Newton’s Second Law: Write the equation $F_{net} = \frac{mv^2}{r}$, where $F_{net}$ is the sum of the forces acting in the radial direction.
4. Solve for the Unknown: Solve the equation for the quantity you are trying to find (e.g., tension, normal force, speed).

Examples

Object on a String

A 0.5 kg mass is attached to a string of length 1.0 m and swung in a vertical circle.

• At the lowest point, the speed is 4.0 m/s. Find the tension in the string.
  $$T - mg = \frac{mv^2}{r}$$
  $$T = mg + \frac{mv^2}{r} = (0.5 \times 9.8) + \frac{0.5 \times 4^2}{1} = 4.9 + 8 = 12.9 N$$

• At the highest point, what is the minimum speed required for the mass to maintain circular motion?
  $$v_{min} = \sqrt{gr} = \sqrt{9.8 \times 1} = 3.13 m/s$$

Object on a Track

A 2.0 kg object moves inside a frictionless vertical circular track of radius 1.5 m.

• At the lowest point, the normal force is 40 N. Find the speed of the object.
  $$N - mg = \frac{mv^2}{r}$$
  $\$40 - (2 \times 9.8) = \frac{2v^2}{1.5}$$
  $\$20.4 = \frac{2v^2}{1.5}$$
  $$v^2 = \frac{20.4 \times 1.5}{2} = 15.3$$
  $$v = \sqrt{15.3} = 3.91 m/s$$

COMMON MISTAKE: Forgetting to consider the direction of the forces (positive or negative) when applying Newton’s Second Law. Always align your coordinate system with the direction of the centripetal acceleration (towards the center of the circle).

Key Equations Summary

STUDY HINT: Practice a variety of problems involving different scenarios (objects on strings, objects on tracks) to solidify your understanding.

Applications

Vertical circular motion principles are applied in:

• Roller Coasters: Analyzing the forces on passengers at the top and bottom of loops.
• Aerobatics: Understanding the forces on pilots during aerial maneuvers.
• Amusement Park Rides: Designing rides that involve circular motion in a vertical plane.

VCAA FOCUS: VCAA often assesses understanding of how forces combine to provide the necessary centripetal force at specific points in the circular path, particularly the highest and lowest points. Expect questions that require you to apply Newton’s Second Law in these situations.