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Wize University Physics Textbook (Master) > Motion in Two Dimensions

Particle in Uniform Circular Motion

Tangential and Radial AccelerationPrevious Section
Centripetal Acceleration and Centripetal ForceNext Section
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Uniform Circular Motion


When an object moves uniformly around a circle, speed of the object remains constant but velocity changes. This is called uniform circular motion (UCM).




Examples of uniform circular motion:
  • A vehicle moving around a circular track
  • Amusement park rides, ferris wheel, merry-go-round
  • The moon, or satellites orbiting the Earth in a circular orbit
  • Any object moving in a circle with constant speed!



Wize Tip
We can model any motion that travels along a circular path at a constant speed with UCM. It does not have to travel the complete circle.

Watch Out!
The object is moving at constant speed (magnitude of velocity), but the object has a changing velocity (the direction constantly changes).

  • Velocity vector is always tangent to the path of the motion. So, for a circular motion, it is tangent to the circle.
  • The change in velocity is caused by an acceleration which is pointing into the centre of the circle and it is known as centripetal acceleration:
a=v2r\boxed{a=\frac{v^2}{r}}
  • Centripetal acceleration is perpendicular to velocity vector at any point




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QUESTION

A ball attached to a 10 cm long string is having a circular motion with period of 2 seconds. If the string makes an angle of 60 degrees with horizontal axis, what is its centripetal acceleration?

SOLUTION

As length of string is 10 cm, the radius of circular motion is:


r=l⋅Cos60=10⋅12=5 cmr=l\cdot Cos60=10\cdot\frac{1}{2}=5\ cm

Period is 2s, hence:


T=2πrvT=\frac{2\pi r}{v}

2=2⋅π⋅5⋅10−2v2=\frac{2\cdot\pi\cdot5\cdot10^{-2}}{v}

v=0.157 msv=0.157\ \frac{m}{s}
Acceleration is then:

ac=v2r=0.225⋅10−2=0.5 ms2a_c=\frac{v^2}{r}=\frac{0.2^2}{5\cdot10^{-2}}=0.5\ \frac{m}{s^2}

A 1000kg car rounds a circular track with radius 10m. If μsmax⁡=0.81,\mu _{s\max }=0.81,what is the maximum speed the car can travel?


A 1.6kg ball is attached to a 1.8m string and is swinging in circular motion horizontally at the string's full length. If the string can withstand a tension force of 87N, what is the maximum speed the ball can travel without the string breaking?


A race car moving with a constant speed of 40m/s completes one lap around a circular track in 50 s. What is the magnitude of the acceleration in m/s2 of the race car?