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.

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.

**Q: What is radial acceleration and how is it related to circular motion?****A:**Radial acceleration is the rate of change of the tangential velocity in a circular motion. It’s always directed towards the center of the circle, and its magnitude is given by $a=v $,/r where $v$ is the tangential velocity, and $r$ is the radius of the circle.**Q: Why is radial acceleration also called centripetal acceleration?****A:**Radial acceleration is called centripetal acceleration because the term “centripetal” means “center-seeking.” This acceleration is directed towards the center of the circular path, describing the nature of the force required to keep an object moving in that path.**Q: How does the radial acceleration change if the radius of the circle is doubled while the speed remains constant?****A:**If the radius is doubled and the speed remains constant, the radial acceleration will be halved, as it’s inversely proportional to the radius.**Q: Can radial acceleration occur in a straight-line motion? Why or why not?****A:**No, radial acceleration specifically refers to the acceleration in a circular motion. It doesn’t apply to straight-line motion because there’s no constant change in direction towards a fixed center point.**Q: If an object is at rest in a circular path, what will be its radial acceleration?****A:**If an object is at rest, its tangential velocity is zero, and consequently, its radial acceleration will also be zero.**Q: How is radial (centripetal) acceleration related to centrifugal force?****A:**Centripetal acceleration is an actual acceleration towards the center of a circular path, whereas centrifugal force is a fictitious force that appears to act outwardly when viewed from a rotating frame of reference. They are equal in magnitude but opposite in direction.**Q: What happens to the radial acceleration if the speed of the object moving in a circular path is doubled?****A:**If the speed is doubled, the radial acceleration will be quadrupled. Radial acceleration is proportional to the square of the velocity, so doubling the velocity increases the acceleration by a factor of four.**Q: What role does friction play in providing radial acceleration for a car taking a turn?****A:**Friction between the tires and the road provides the centripetal force required for radial acceleration. Without sufficient friction, the car would not be able to change direction and maintain a circular path, and would instead continue in a straight line.**Q: Is it possible for radial acceleration to be negative? Why or why not?****A:**Radial acceleration is always directed towards the center of the circle, so it’s defined as positive in that direction. It cannot be negative since the direction of radial acceleration is by definition towards the center of the circle.**Q: How does gravity contribute to radial acceleration in the case of celestial objects like planets orbiting the sun?****A:**In the case of planets orbiting the sun, the gravitational force between the two bodies acts as the centripetal force, providing the radial acceleration needed to keep the planet in its circular (or nearly circular) orbit. The gravitational force keeps the planet moving in a path around the sun, rather than moving off in a straight line.

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