A particle moves in a circle of radius 25 cm at two revolutions per second. The average acceleration of the particle in half cycle in meters per sec² is:
(A) π
(B) 8π
(C) 800π
(D) 2π
Answer (1)
To find the average acceleration of a particle moving in a circle for half a cycle, we need to consider the change in velocity over time.
Given:
Radius of the circle, r = 25 cm = 0.25 m
Frequency of rotation, f = 2 rev/s
Calculations:
Determine the initial and final velocities:
The particle completes one half cycle, moving from diametrically opposite points on the circle. In such cases, the initial and final velocities are equal in magnitude but opposite in direction.
The linear velocity v is given by: v = 2 π r f v = 2 π × 0.25 × 2 = π m/s
Initial velocity v i = π m/s in one direction.
Final velocity v f = − π m/s in the opposite direction.
Calculate the change in velocity:
The change in velocity Δ v is: Δ v = v f − v i = − π − ( π ) = − 2 π m/s
Time for half a cycle:
The time for half a cycle, T = 2 f 1 = 4 1 seconds .
Calculate the average acceleration:
The average acceleration a ˉ is given by the change in velocity over time: a ˉ = T Δ v = 4 1 − 2 π = − 8 π m/s 2
Since acceleration is direction-sensitive, the magnitude of average acceleration is 8 π m/s 2 .
Thus, the correct answer is (B) 8\pi .
