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Wize University Physics Textbook (Master) > Circular Motion

Uniform Circular Motion

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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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Period and Frequency

  • Period (T)= the time it takes for the object to make one complete revolution or cycle (measured in seconds)
  • Frequency (f)= the number of cycles made in a given time period. Usually it is expressed per second. (measured in Hz where 1 Hz = 1/s).
  • Period and frequency are inversely related as follows:
f=1T\boxed{f=\dfrac{1}{T}}

  • Circumference of a circle of radius rr is 2πr2\pi r.So, time to complete one revolution is:
T=2πrv\boxed{T=\frac{2\pi r}{v}}

  • Angular velocity is defined as the rate at which angles are changing in a circular motion and it could be found as:
ω=2πT\boxed{\omega=\frac{2\pi}{T}}


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Example: A Circulating Ball

A ball attached to a 10 cm long string is having a circular motion with period of 0.59 seconds. If the string makes an angle of 60 degrees with horizontal axis, what is its centripetal acceleration?

Solution:


Note that the radius of circular path in which the ball is moving is different from the length of the string. However, we can find this radius by a little bit of trigonometry knowing the angle the string makes with the horizontal direction.
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
Now we can use the period and the radius of this motion to find the speed of the ball:
Period is 2s, hence:


T=2πrvT=\frac{2\pi r}{v}

0.59=2⋅π⋅5⋅10−2v0.59=\frac{2\cdot\pi\cdot5\cdot10^{-2}}{v}

v=0.532 msv=0.532\ \frac{m}{s}
Acceleration is then:

ac=v2r=0.53225⋅10−2=5.66 ms2a_c=\frac{v^2}{r}=\frac{0.532^2}{5\cdot10^{-2}}=5.66\ \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?


Extra Practice
Example : circular motion
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 car goes around a curve of radius r at a constant speed v. Then it goes around the same curve at half of the original speed. What is the centripetal force on the car as it goes around the curve for the second time, compared to the first time?
Practice question
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?
Practice question
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?
Example : circular motion
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?
Example : circular motion
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?
Example : circular motion
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 1000 kg car rounds a flat, circular track with radius 10 m. If μsmax⁡=0.81,\mu _{s\max }=0.81,what is the maximum speed the car can travel without slipping? (in m/s, two sig figs)
Roller coaster - vertical circular motion
A roller coaster vehicle has a mass of 500 kg when fully loaded with passengers. (a) If the vehicle has a speed of 20.0 m/s at point A, what is the force of the track on the vehicle at the point? (b) What is the maximum speed the vehicle can have at point B for gravity to hold it on the track?
A boy standing in one spot, swings a box on a string around him as fast as he can. The box ends up moving around him in a circle at a constant height. The string is 1.25 meters long.
a) Draw a free-body diagram for the box. (Hints: what are the forces in the vertical direction? What about the horizontal direction? Remember: centripetal force is NOT some "new" force, it is included in the forces already on your FBD!))
b) If the tension in the string is 100 N and the mass of the box is 5.0 kg, how fast is the box moving? (That is, what is the tangential speed? Enter your answer to two sig figs in meters per second.)
A carousel moves at 8.0m/s. The radius of the circle is 7.4m.
A 76kg pilot does a vertical loop in his plane. The loop is 52.0m in radius, and at the bottom it is moving at 48m/s. What is the normal force on the pilot?
A bug is sitting on a spinning disk. The bug is slowing moving into the centre of the disk. Which of the following statements are true?
A man riding a bicycle with a 100 kg total mass is performing horizontal circles on flat track. If the coefficient of static friction between the tires and track is 0.600, find the smallest radius the man can make when travelling at a speed of 10.0 m/s.
Ellen is swinging a 0.01kg yo-yo in a circular path perpendicular to the ground. The yo-yo moves in a clockwise direction with a constant speed of 2ms.
What is the velocity of the yo-yo at the bottom of the circle?
Example : circular motion
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 car goes around a curve of radius r at a constant speed v. Then it goes around the same curve at half of the original speed. What is the centripetal force on the car as it goes around the curve for the second time, compared to the first time?
Uniform Circular Motion
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 ball of mass M is released from rest at a vertical height h above the ground. The ball rolls down a frictionless inclined plane and makes it to a loop the loop of radius R.
a) What is the velocity of the ball as it reaches to the bottom of the incline?
b) What is the normal force at the bottom?
An object in a uniform circular motion (constant speed v around a circle with constant R is acted upon by a centripetal force. Which statement below is FALSE? The centripetal force:
A runner is running at a speed of 10m/s along a circular track of radius 25 m.
(a) Find the time it takes to complete one lap when he is running at constant speed.
(b) Find the radial acceleration during this time.
Practice question
A race car moving with a constant speed of 40m/s completes one lap around a circular track in 50s. What is the magnitude of the acceleration in m/s2 of the race car?
Circular Motion
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?