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Activity: Circular motion

Circular motion, like linear motion, has a number of quantities to be defined, described and derived, including centripetal acceleration. Here you will also understand that a constant force perpendicular to direction of motion results in a circular path and become able to solve problems involving motion in a circle.

Need to know


Doing this at home?

All of this is OK for work at home (perhaps even with a few objects on strings or shoelaces!) except that you will have to get yourself a pig for the Optional Practical: Flying Pig. It does, of course, have to be a pig. Cows and dinosaurs are not good enough and birds simply won’t do.

To save you having to invest in a pig, here is a video of me doing it. You can use LoggePro video analysis to measure the vertial distance from the top of the string to the pig. To get it into LoggerPro you could screenshot the paused video for each length and use the still images

Theory

Motion of a ball on a string

In this example we will consider the well known example of the motion of a ball moving in a circle on the end of a string, however to make the forces simpler we will consider this in space.

 To describe this motion we will define some new terms:

Time period (T) = time for one complete circle - unit: s

Frequency (f) = number of complete circles per second (1/T) - unit: Hz

Angular displacement (θ) = angle swept out in radians - unit: rad

Angular velocity (ω) = angle swept out per unit time - unit: rad s-1

Radians

Speed

For the animation below...

  • Measure the time period
  • Calculate the frequency
  • What is the angle swept out after 1 revolution?
  • Calculate the angular velocity
  • What is the distance travelled after 1 revolution?
  • Calculate the speed.

Note the following equation:

s p e e d space equals space fraction numerator 2 pi r over denominator T end fraction space equals space omega r

Small angle approximation

An angle in radians = arc length/radius (s/r) But if the angle is small this is approximately the same as the chord length/radius.

  • Vary the angle by moving the blue dot and see why the approximation is only good for small angles.

Centripetal acceleration

  • Use the small angle approximation to write an expression for angle Δθ
  • Rearrange to make Δv the subject
  • divide by Δt to give an equation for the acceleration and show that:

    a equals v omega equals omega squared r equals v squared over r

You can see from the vector addition that the direction of the acceleration is towards the centre hence the name centripetal.

It looks like the change in velocity is zero but you have to remember velocity is a vector so you must subtract the vectors.

Centripetal force

Applying Newton's second law gives:

F equals m omega squared r equals fraction numerator m v squared over denominator r end fraction

Examples

Mass on a string (no gravity)

  • What force holds the ball in circular motion?
  • If the ball has a mass 2 kg and the radius is 1m estimate the centripetal force.
  • What would happen if the string were cut?

Mass on a string with gravity (vertical)

When a ball moves in a vertical circle its speed is not constant.

  • At what point is the ball travelling slowest?
  • Use the conservation of energy to explain why the ball slows down.
  • Draw a free body diagram showing the forces on the ball at the bottom of the circle.
  • use newton's 2nd law to write an expression for the Tension at the bottom.
  • Draw a free body diagram showing the forces on the ball at the top of the circle.
  • Use newton's 2nd law to write an expression for the Tension at the top.

The animation below shows a ball travelling around a vertical circle at minimum velocity:

  • What is the tension at the top?
  • Write an expression for the minimum velocity at the top.
  • Calculate the minimum velocity for a 2 kg ball to complete a circle of radius 1 m.

Mass on a string with gravity (horizontal)

To complete a horizontal circle the string must be at an angle.

  • Draw a free body diagram showing the forces acting on the ball.
  • Which force provides the centripetal force?

You can experiment with looping the loop using the PhET energy park simulation.

And now, the moment you've all been waiting for. The Flying Pig practical!

Post a screenshot of your graph and write the value of g obtained.

What could be done to improve the result?

Words: 1-200

 

 


You can test your GeoGebra skills by building a simulation of the flying pig Pig simulation.

Explain how the following examples move in a circle

Bob Burnquist looping the loop

The wall of death

Summary

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