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Wize High School Grade 12 Physics Textbook > Circular Motion

Centripetal Acceleration and Centripetal Force

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Centripetal Acceleration and Centripetal Force


In circular motions, there is a centripetal acceleration (pointed toward centre of circle) changing the direction of velocity. If there's acceleration, there must be a force! We call this a centripetal force.


Wize Concept
This force is not a separate force. The centripetal force is the name we give to whatever force is already there and resulting in circular motion. For example, it could be friction, an applied force, gravity, or tension. It could also be a component of any of these forces.

For an object of mass mm moving at speed of vvaround a circular path of radius rr:

The magnitude of this centripetal acceleration is given by:

ac=v2r\boxed{a_c=\frac{v^2}{r}}

Thus using the second law of Newton, the centripetal force has magnitude of:

Fc=mac=mv2r\boxed{F_c=ma_c=\frac{mv^2}{r}}

Watch Out!
Note that both centripetal acceleration and centripetal force are pointing toward the center of the circular path.


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Tips for solving uniform circular motion problems:
  • Draw a clearly labeled free body diagram.
  • Calculate the net force
  • Set the net force equal to the centripetal force equation (net force = centripetal force here)
  • Solve for the unknown!

Examples:

Mass on a string
  • The centripetal force here is provided by the tension
  • If the string is swung horizontally, the tension will be equal to the centripetal force.
  • If the string is held at an angle, the centripetal force will be the component of tension in towards the center of the circle.



Vertical Circular Motion
  • Remember, the centripetal force is the net force towards the centre of the circle.
  • Often here, it will be the sum of the force of gravity and another force acting on the object (ie. tension or a normal force).
  • Choose the direction towards the centre of the circle to be positive.
  • This means that the signs of forces will depend on where the object is located in the circle!
  • Make sure you read the question carefully and set up the free body diagram accordingly.



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Example: Circular Paths with a Different Radius


A bobsled track consists of two turns with radii 54 and 36 meter as shown in the picture. If bobsled turns with constant speed v,v, what is ratio of centripetal force f2/f1f_2/f_1, acting on the bobsled while turning around the two turns?

A) 1.9 B) 1.2 C) 0.66 D) 1.5



Solution:

Fc1=mv2r1 and  Fc2=mv2r2 ⇒  Fc2Fc1 =r1r2=1.5F_{c1}=\frac{mv^2}{r_1}\ and\ \ F_{c2}=\frac{mv^2}{r_2}\ \Rightarrow\ \ \frac{F_{c2}}{F_{c1}}\ =\frac{r_1}{r_2}=1.5




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Practice: Ball on String at an Angle

A 2.0 kg ball is suspended by a 1.20 m long string. A girl grabs the string and swings it downwards in front of her at a constant speed. The string makes an angle of 35º with the vertical. Determine the speed of the ball.

Practice: Ferris Wheel

A 65 kg man is riding a Ferris wheel with radius of 15 m, that moves at speed 2.355 m/s. Determine the normal force that the man experiences at the top of the ferris wheel and at the bottom.
A box of mass 2.50 kg is attached to a string and whirled in a vertical circle of radius 1.0 m. At the exact top of the path, the tension in the string is 3 times the ball’s weight.


What is the ball’s speed at this point?
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Practice: Friction Turntable

Three blocks are placed on a spinning turntable at increasing distances away from the center of the turntable. Recall that the tangential velocity increases towards the edge of the turntable. The coefficient of static friction for all three blocks is the same, and they have the same mass. As the velocity of the turntable increases, which block slips first? Why?

A 20-g marble is free to slide on a frictionless wire loop with radius of 15 cm. The loop starts to rotate about its vertical axis until the bead slides up to θ=60∘ as shown below. What is the angular velocity of the loop at this moment?



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