Question

Object A and Object B are in uniform circular motion. If the linear velocities of both objects are equal, and the radius of the motion of Object. is 4 times the radius of the motion of Object A, the radial acceleration of Object A is how many times the radial acceleration of Object B?

Answer 1

In uniform circular motion, the velocity is tangential and the acceleration is radially outward. The centripetal acceleration can be calculated using the formula ac = ω²r. In this scenario, the radial acceleration of Object A is 16 times larger than that of Object B.

Step-by-step explanation:

When objects are in uniform circular motion, the velocity of the object is tangential to the circular path and the acceleration is radially outward, towards the center of the circle.

The centripetal acceleration, ac, can be calculated using the equation ac = ω²r, where ω is the angular velocity and r is the radius of the circular path.

In this scenario, Object A has a larger radius than Object B, but both objects have the same linear velocity. Since the radius of Object A is 4 times the radius of Object B, the radial acceleration of Object A would be 4² = 16 times larger than the radial acceleration of Object B.

Answer 2

The radial acceleration of object A is 4 times that of object B.

Step-by-step explanation:

The radial acceleration, also known as centripetal acceleration, is given by

`a = (v^2)/(r)`

v represents the linear velocity and r the radius of motion.

For object A, let its radial acceleration be
`a_A`, its linear velocity be v and its radius of motion be r. Then

`a_A = (v^2)/(r)`

For object B, let its radial acceleration be
`a_B`. It has the same linear velocity, v, as object A and its radius of motion is 4 times that of object A, 4r. Then

`a_B = (v^2)/(4r) = (1)/(4)\left((v^2)/(r)\right)`

But the expression in parentheses is
`a_A`.

`\therefore a_b = (a_A)/(4)`

`a_A = 4a_B`

Hence, the radial acceleration of object A is 4 times that of object B.