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

Tangential and Radial Acceleration

The Position, Velocity, and Acceleration VectorsPrevious Section
Particle in Uniform Circular MotionNext 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.


  • Speed of the object is found by dividing circumference of the circle (2πr)(2\pi r) by the period of motion T (time to traverse a full circle):
v=2πrTv=\frac{2\pi r}T


  • Instantaneous velocity here is tangent to the circular path at any point.

  • In uniform circular motion, Δv\Delta v vector which shows the direction of acceleration, is pointing towards the center of rotation.
The acceleration is called Centripetal Acceleration- directed towards the center of a circle, that keeps an object moving in a circular path.
ac=v2r=rω2a_c=\frac{v^2}{r}=r\omega^2
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!




Centripetal Force

  • For objects moving in uniform circular motion, the acceleration always points in towards the center of the circle.
  • This means that there must be a net force towards the center of the circle that is providing this acceleration.
This force is not a separate force, but it is a force that is part of the situation acting on the object or is a net force. For example, it could be friction, an applied force, gravity, or tension.


  • The centripetal force is equal to the net force on the object. This is the condition in which motion in a circle enforces.
  • For an object of mass m moving at tangential speed v around a circular path of radius r, the magnitude of its centripetal force is:


Fc=mac=mv2rF_c=ma_c=\frac{mv^2}r
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!
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  • Non-uniform circular motion still has a circular path, but the speed of the object is not constant.
  • For this motion, acceleration is not solely centripetal (radial), but a tangential component shows up too.
  • Tangential acceleration is defined as:
at=dvtdt=d(rω)dt=rdωdt=rαa_{t}=\frac{dv_{t}}{dt}=\frac{d(r\omega)}{dt}=r\frac{d\omega}{dt}=r\alpha

where the magnitude of the acceleration is a=ac2+at2a=\sqrt{a^2_c+a^2_t}.
at is tangential acceleration
ac is centripetal (or radial) acceleration
  • The tangential component is responsible for the change in speed, while the radial component remains responsible for the circular motion.
  • The net acceleration does not point towards the center of the circle in non-uniform circular motion.




Note:
  • If the angle between at and v is less than 90o, the object is speeding up
  • If the angle between at and v is greater than 90o, the object is slowing up
An object rotates at distance of 1.8m from the center, traveling with an angular velocity of 2.0 rad/s. What is the centripetal acceleration of the object?

Centripetal (aka Radial) Acceleration = ac = ar = v2/r

Linear (v) and Angular Velocity (w): v = r w

therefore, ac = (r w)2/r = r w2 = (1.8)(2.0)2 = 7.2 m/s2
A car travels in a circular track of radius 10 m. Its speed is changing at a rate of 15.0 m/s2 at an instant when its speed is 40.0 m/s. What is the magnitude of the acceleration of the car?