# Radial acceleration – problems and solutions

1. Which graph below shows the relation between centripetal acceleration or radial acceleration (a_{R}) and linear velocity (v) in uniform circular motion.

Solution :

The equation of the radial acceleration :

*a*_{R}* = **radial acceleration**, v = **linear velocity**, r = **distance from the axis of rotation.*

We investigate the relation between the radial acceleration (a_{R}) with the linear velocity (v) so that distance from the axis of rotation (r) is constant. For example, r = 1.

Graph A.

2. A ball rotates in a container with a diameter of 1 meter. If the angular speed is 50 rpm, what are the linear velocity and radial acceleration of the ball?

__Known :__

Diameter of circle (D) = 1 m

Radius of circle (r) = 0,5 m

Angular speed (ω) = 50 rpm = 50 revolutions / 1 minute

1 revolution = 2π radian

50 revolutions = 50 (2π radian) = 100π radian

1 minute = 60 seconds

Angular velocity (ω) = 100π radian / 60 seconds = (10π/6) radian/second

__Wanted :__ Linear speed (v) and radial acceleration (a_{R})

__Solution :__

Linear speed (v) :

v = r ω = (0.5)(10π/6) = 5π/6 m/s

Radial acceleration (a_{R}) :

a_{R} = v^{2}/r = (5π/6)^{2 }: 0.5 = 25π^{2}/36 : 0.5 = (25π^{2}/36)(1/0.5)

a_{R} = (25π^{2}/18) m/s^{2}

3. An object travels at a constant speed v in a circle with radius of R and radial acceleration a_{R}. If the radial acceleration becomes 2 times, then v becomes ………. times and radius becomes ………. times

Solution :

The equation of the radial acceleration :

If the radial acceleration (a_{R}) = 1 then the linear speed (v) = 1 and radius (r) = 1 :

If the radial acceleration (a_{R}) = 2 then the linear speed (v) = 2 and radius (r) = 2 :

If the radial acceleration becomes 2 times, then the linear speed (v) becomes 2 times and the radius of circle becomes 2 times.