A 56 cm diameter wheel accelerates uniformly about its center from 130 rpm to 400 rpm in 3.2 s.
(a) Determine its angular acceleration.
Express your answer using two significant figures.
(b) Determine the radial component of the linear acceleration of a point on the edge of the wheel 1.0 s after it has started accelerating.
(c) Determine the tangential component of the linear acceleration of a point on the edge of the wheel 1.0 s after it has started accelerating.
To solve these problems, we will use the following equations:
(a) Angular acceleration (α) = (ωf - ωi) / t
(b) Radial component of linear acceleration (ar) = r * α
(c) Tangential component of linear acceleration (at) = r * α
Given information:
- Diameter of the wheel (d) = 56 cm = 0.56 m
- Initial angular velocity (ωi) = 130 rpm = 130 * 2π rad/min
- Final angular velocity (ωf) = 400 rpm = 400 * 2π rad/min
- Time (t) = 3.2 s
- Time after starting acceleration (t') = 1.0 s
- Radius (r) = d/2 = 0.56/2 = 0.28 m
(a) Angular acceleration (α) = (ωf - ωi) / t
= (400 * 2π rad/min - 130 * 2π rad/min) / 3.2 s
= (270 * 2π rad/min) / 3.2 s
≈ 530 rad/min^2
Therefore, the angular acceleration is approximately 530 rad/min^2.
(b) Radial component of linear acceleration (ar) = r * α
= 0.28 m * 530 rad/min^2
≈ 150 m/s^2
Therefore, the radial component of the linear acceleration of a point on the edge of the wheel 1.0 s after it has started accelerating is approximately 150 m/s^2.
(c) Tangential component of linear acceleration (at) = r * α
= 0.28 m * 530 rad/min^2
≈ 150 m/s^2
Therefore, the tangential component of the linear acceleration of a point on the edge of the wheel 1.0 s after it has started accelerating is approximately 150 m/s^2.