A rotor 0.4 m in diameter is given a constant angular acceleration from rest so that it makes 50 revolutions in 10 seconds. what is the tangential acceleration for a point on its rim? (b) what is the angular acceleration of the rotor?
To find the tangential acceleration for a point on the rim, we need to determine the linear speed of the point and the time it takes to reach that speed.
To calculate the linear speed, we need to find the distance (circumference) covered by a point on the rim during 50 revolutions. The circumference (C) of a circle is given by the formula:
C = 2πr
where r is the radius of the circle. In this case, the radius is half of the diameter, so r = 0.4 m / 2 = 0.2 m.
C = 2π * 0.2 = 0.4π m
Since the point makes 50 revolutions, the distance covered is 50 times the circumference:
distance = 50 * C = 50 * 0.4π ≈ 62.83 m
Next, we need to determine the time it takes for the point to travel this distance. We are given that this time is 10 seconds.
Now, we can calculate the linear speed (v) using the formula:
v = distance / time
v = 62.83 m / 10 s ≈ 6.283 m/s
The tangential acceleration (a_t) is the rate at which the linear speed of the point changes. Since the rotor starts from rest, the initial linear speed is 0 m/s.
Using the formula for tangential acceleration:
a_t = (final speed - initial speed) / time
a_t = (6.283 m/s - 0 m/s) / 10 s = 0.6283 m/s^2
Therefore, the tangential acceleration for a point on the rim is approximately 0.6283 m/s^2.
To find the angular acceleration of the rotor, we can use the formula:
Angular acceleration (α) = (final angular velocity (ω) - initial angular velocity (ω_0)) / time
We are given that the rotor starts from rest, so the initial angular velocity is 0 rad/s.
The final angular velocity is the angular displacement (θ) divided by the time taken. The angular displacement can be calculated by converting the number of revolutions (50) into radians.
One revolution is equal to 2π radians.
θ = 50 revolutions * 2π radians/revolution = 100π radians
Therefore, the final angular velocity is:
ω = θ / time = 100π radians / 10 s = 10π rad/s
Now we can calculate the angular acceleration:
α = (10π rad/s - 0 rad/s) / 10 s = π rad/s^2
Therefore, the angular acceleration of the rotor is π rad/s^2.