# 6.11 Centripetal Force

## Centripetal Force

•  Centripetal force can be calculated using any of the following equations:

• Where:
• F = centripetal force (N)
• v = linear velocity (m s-1)
• ⍵ = angular speed (rad s-1)
• r = radius of the orbit (m)

Centripetal force is always perpendicular to the direction of travel

• The centripetal force is the resultant force on the object moving in a circle
• This is particularly important if there are multiple forces on the object, such as weight

#### Vertical Circular Motion

• An example of vertical circular motion is swinging a ball on a string in a vertical circle
• The forces acting on the ball are:
• The tension in the string
• The weight of the ball downwards

• As the ball moves around the circle, the direction of the tension will change continuously
• The magnitude of the tension will also vary continuously, reaching a maximum value at the bottom and a minimum value at the top
• This is because the direction of the weight of the ball never changes, so the resultant force will vary depending on the position of the ball in the circle

• At the bottom of the circle, the tension must overcome the weight, this can be written as:

• As a result, the acceleration, and hence, the speed of the ball will be slower at the top
• At the top of the circle, the tension and weight act in the same direction, this can be written as:

• As a result, the acceleration, and hence, the speed of the ball will be faster at the bottom

#### Worked example

A bucket of mass 8.0 kg is filled with water and is attached to a string of length 0.5 m.

What is the minimum speed the bucket must have at the top of the circle so no water spills out?

Step 1: Draw the forces on the bucket at the top

Step 2: Write an expression for the centripetal force

• The weight of the bucket = mg
• At the top of the circular path, the weight and tension act in the same direction, so the centripetal force is

• The minimum speed v is when the string is taut but not stretched, so the tension here is zero (T = 0)

Step 3: Rearrange for velocity v and calculate

• m cancels from both sides

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