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.
a. 130 REV/min * 1/60 min/s = 2.17 REV/S.
400 REV/min * 1/60 min/s = 66.7 REV/s.
a = (Vt - Vo) / t,
a = (66.7 - 2.17) / 3.2 = 20.2 REV/S^2.
a = 20.2 REV/S^2 * 2piRAD/REV,
= 40.4piRAD/s^2 = 126.9 RAD/s^2.
(a) To determine the angular acceleration of the wheel, we can use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given that the initial angular velocity is 130 rpm and the final angular velocity is 400 rpm, we need to convert these to rad/s. Recall that 1 rpm = π/30 rad/s. Therefore:
initial angular velocity = 130 rpm * π/30 rad/s = 13π/3 rad/s
final angular velocity = 400 rpm * π/30 rad/s = 40π/3 rad/s
Using the formula for angular acceleration:
α = (40π/3 - 13π/3) / 3.2 s ≈ 8.94 rad/s²
Therefore, the angular acceleration of the wheel is approximately 8.94 rad/s².
(b) To determine the radial component of the linear acceleration of a point on the edge of the wheel, we can use the formula:
linear acceleration (a) = radius of the wheel * angular acceleration
The radius of the wheel is half of its diameter, which is 56 cm / 2 = 28 cm = 0.28 m.
Using the angular acceleration value calculated in part (a):
a = 0.28 m * 8.94 rad/s² ≈ 2.50 m/s²
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 2.50 m/s².
(c) The tangential component of the linear acceleration can be calculated using the formula:
tangential acceleration (at) = radius of the wheel * angular velocity²
Using the radius of the wheel:
at = 0.28 m * (40π/3 rad/s)² ≈ 74.20 m/s²
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 74.20 m/s².
(a) To determine the angular acceleration, we can use the formula:
angular acceleration = (final angular velocity - initial angular velocity) / time
Given:
Initial angular velocity = 130 rpm
Final angular velocity = 400 rpm
Time = 3.2 s
First, let's convert the angular velocities to radians per second:
1 revolution = 2π radians
1 minute = 60 seconds
Initial angular velocity = (130 rpm) * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Final angular velocity = (400 rpm) * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Now we can calculate the angular acceleration:
angular acceleration = (Final angular velocity - Initial angular velocity) / time
(b) To 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, we can use the formula:
radial acceleration = (angular acceleration) * (radius)
Given:
Radius = 56 cm
(c) To 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, we can use the formula:
tangential acceleration = (angular acceleration) * (radius)
Now let's plug in the values and solve for each part.