Circular Motion

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We all know what a circle is, but what physics principles govern an object moving in a circular path?

How do we analyze and describe its motion as it travels in a circle? In this series of lessons on kinematics and dynamics, we will explore:

What circular motion is and how it is defined
How circular motion is measured mathematically
Real-world applications and complex examples
Centripetal force and the underlying causes of circular motion

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Before you start, it would be helpful to have reviewed linear kinematics and the concept of force

Understanding Circular Motion

In this lesson we will examine the principles behind uniform circular motion. In other words, situations when the object undergoing circular motion is traveling at a constant speed.
The path has a constant radius (r) and a Period (T).
However, the velocity is not constant.

Although the magnitude of the velocity (which is the speed) is constant, its direction is constantly changing, since it is a vector.

This leads to an important point.
Since the velocity is changing, it must be accelerating.
Since it is accelerating, it must be experiencing a net force.
It is this force, the centripetal force, that causes the object to undergoing circular motion.

The following video explores this in greater detail, including the formulas involved in the analysis of circular motion

Check your understanding

Multiple Choice Practice Questions

1. An object is moving at a constant speed in a horizontal circle. Which of the following statements is true regarding its velocity and acceleration?

A) Both velocity and acceleration are constant. B) The velocity is constant, but the acceleration is changing. C) The magnitude of the velocity is constant, but its direction is changing, causing an acceleration directed toward the center. D) The acceleration is directed tangentially to the path of motion.

2. If a car travels around a circular track at a constant speed, what happens to the required centripetal force if the car's speed is doubled while maintaining the same radius?

A) The centripetal force remains the same. B) The centripetal force doubles. C) The centripetal force is halved. D) The centripetal force increases by a factor of 4.

3. A ball attached to a string is swung in a clockwise horizontal circle. If the string breaks suddenly when the ball is at the top of the circle (12 o'clock position), in which direction will the ball travel immediately after the break?

A) Straight to the right (tangent to the circle at 12 o'clock). B) Radially outward away from the center. C) Radially inward toward the center. D) It continues moving in a circular arc for a few seconds before going straight.

Sample Problem
We are now ready to try some sample problems
Below are sample problems with videos that explain how to solve them. It is suggested you try the problems beforehand, as this actually aids understanding, even if you are unsure if you are correct.

Here are some more problems to try out

1. An ant is riding on the tip of a 10 cm long second hand of a clock. Find the linear velocity of the ant.

Given: Radius r = 10 cm = 0.10 m, Period T = 60 s (second hand takes 60 s per full rotation)

Formula: v = (2 π r) / T

Calculation: v = (2 × π × 10 cm) / 60 s ≈ 1.047 cm/s (or 0.0105 m/s)

Answer: 1.05 cm/s (or 0.62 cm/s depending on reference framing)

2. Calculate the centripetal force required to keep a 5 kg mass revolving in a circular path of radius 125 cm with a period of two seconds.

Given: m = 5 kg, r = 125 cm = 1.25 m, T = 2 s

Step 1 (Find Velocity): v = (2 π r) / T = (2 × π × 1.25) / 2 = 1.25 π ≈ 3.927 m/s

Step 2 (Centripetal Force): F_c = (m v²) / r = m × (4 π² r) / T²

F_c = 5 × (4 × π² × 1.25) / (2²) = 5 × π² × 1.25 ≈ 61.685 N

Answer: 61.9 N

3. An astronaut is on a spinning space station so that he experiences artificial gravity, just like in the film 2001: A Space Odyssey. If the radius of the station is 1000 m, what must be the tangential velocity to simulate Earth's gravity (g = 9.8 m/s²)?

Given: Radius r = 1000 m, Centripetal Acceleration a_c = g = 9.8 m/s²

Formula: a_c = v² / r → v = √(a_c × r)

Calculation: v = √(9.8 × 1000) = √(9800) ≈ 98.99 m/s

Answer: 99 m/s

4. As you go around a corner at 72 km/h, you experience a force of 1120 N from the seat of your car. If your mass is 70 kg, what is the radius of the curve?

Given: Speed v = 72 km/h = 72 / 3.6 = 20 m/s, F_c = 1120 N, m = 70 kg

Formula: F_c = (m v²) / r → r = (m v²) / F_c

Calculation: r = (70 × 20²) / 1120 = (70 × 400) / 1120 = 28000 / 1120 = 25 m

Answer: 25 m

5. A centripetal force of 8.94 N is required to keep an object in a circular orbit. What is the mass of the object if it is travelling with a constant speed of 8.10 m/s and a radius of 5.29 m?

Given: F_c = 8.94 N, v = 8.10 m/s, r = 5.29 m

Formula: F_c = (m v²) / r → m = (F_c × r) / v²

Calculation: m = (8.94 × 5.29) / (8.10²) = 47.2926 / 65.61 ≈ 0.7208 kg

Answer: 0.72 kg

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Understanding Circular Motion

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