Calculating Circular Motion: Questions & Answers

From the previous article we looked at circular motion, its definition, formulas involved, types of velocities etc. Now we will look at different calculations involving circular motion. This tutorial is intended to give you a well-rounded understanding on how to calculate different variables involved in circular motion questions namely:

• Calculating Tension.
• Calculating Centripetal force.
• Calculating centripetal acceleration.
• Calculating scalar and angular velocities.
• And calculating period and frequency.

Let’s first begin with some short descriptions and formulas.
When an object is travelling in a circular motion its direction is changing continually, this results in a change in velocity it also follows that any change in velocity causes acceleration and for an acceleration to take place a force is needed. The force present when a body is moving in a circular motion is known as centripetal force, this force acts towards the center of motion.

It is also said that the centripetal acceleration must move in the same direction as the centripetal force, therefore it also acts toward the center of motion. The formulas below show the relationship between centripetal acceleration and angular and scalar velocity:

To read more about angular and scalar velocity click here.

Example #1

The pendulum above has a mass of 2kg. The pendulum is moving in a horizontal circle. The string is inclined at 30 degrees to the vertical.

1. Calculate the tension in the string
2. Calculate the centripetal force.
3. Calculate the centripetal acceleration.
4. Calculate Scalar & angular velocities.
5. Calculate period & frequency of centripetal motion.

Answers:
First before we begin answering the question we need to make some additions to the drawing above. We want to work with an angle besides the 30 degrees if you look at the diagram you’ll realize there’s a triangle but one of the angles are missing, this is the angle we want to use. You can easily find this angle because you are already given one angle (30 degrees) and since it is a right angled triangle (indicated by the box formed between the vertical and horizontal line)it also has 90 degrees, therefore the remaining angle must be 60 degrees, if we were to add them all up we would get a total of 180 degrees. So place the 60 degrees in the missing position. The question also stated that you would need to calculate tension, so you would need to split the “T” in its horizontal and vertical components. This would give “T × sin 60” for the vertical component and “T × cos 60” for the horizontal component. Now place these components in the diagram. The next thing to note is that the pendulum has a mass 2kg therefore a downward force of weight must be acting on the body, add this as well to the diagram.

The resulting diagram would be:

1. To answer the first question which asked for the tension in the string we use the vertical component but if you look closely you will see that there is weight also acting downward against the vertical component so we use the rule:

2. Calculating centripetal force. Remember as stated above that this is the force acting toward the center. Therefore if you look at the diagram you will realize that the horizontal component ( T × cos 60)is the one acting toward the center. This would give:


Centripetal force = T × cos 60 
= 23.9 × 0.5 
= 11.55N


3. Calculating centripetal acceleration. Remember this moves in the same direction of the force and you’re also given mass so all you have to do is transpose this formula to find acceleration:





4. In this question they asked for the scalar and angular velocities. The formula that relates these to centripetal acceleration was given above:


Where r is the radius

Since we weren’t given the value of the radius we would have to find it first before we can find either scalar or angular velocity. Assuming we all can identify the radius on the diagram let’s use the formula:



If you look at the triangle formed on the circle you would notice that the adjacent would be the radius and the hypotenuse would be the 1.2m. Therefore:

Calculating scalar velocity:

Calculating angular velocity:

5. Calculating period:

Calculating frequency: