At time t=0 a grinding wheel has an angular velocity of 26.0 rad/s. It has a constant angular acceleration of 25.0 rad/s^2 until a circuit breaker trips at time t = 2.50 s. From then on, the wheel turns through an angle of 440 rad as it coasts to a stop at constant angular deceleration.
Through what total angle did the wheel turn between t=0 and the time it stopped? At what time does the wheel stop? What was the wheel's angular acceleration as it slowed down?
To find the total angle through which the wheel turned between t=0 and the time it stopped, we need to find the angle covered during the period of constant angular acceleration and the angle covered during the period of constant angular deceleration.
1. Angle covered during the period of constant angular acceleration:
Using the formula for angular displacement (θ) with constant angular acceleration (α):
θ = ω_i * t + 0.5 * α * t^2
where:
ω_i = initial angular velocity
t = time interval
Substituting the given values:
ω_i = 26.0 rad/s
t = 2.50 s
α = 25.0 rad/s^2
θ_acc = (26.0 rad/s) * (2.50 s) + 0.5 * (25.0 rad/s^2) * (2.50 s)^2
Calculate θ_acc to find the angle covered during this period.
2. Angle covered during the period of constant angular deceleration:
The final angular velocity (ω_f) at the time the circuit breaker trips can be found using the equation:
ω_f = ω_i + α * t
By substituting the values:
ω_i = 26.0 rad/s
α = 25.0 rad/s^2
t = 2.50 s
Calculate ω_f.
Next, use the formula for angular displacement with constant angular deceleration (α_deceleration):
θ_deceleration = ω_f * t + 0.5 * α_deceleration * t^2
The unknown value here is α_deceleration. We will solve for this value in the next step.
3. Angular acceleration as it slowed down:
The given information states that the wheel coasts to a stop at constant angular deceleration. This implies that the final angular velocity (ω_f) at the time the wheel stops is 0 rad/s.
Using the equation:
ω_f = ω_i + α_deceleration * t_deceleration,
where t_deceleration is the time it takes for the wheel to stop.
Rearranging the equation to solve for α_deceleration:
α_deceleration = (ω_f - ω_i) / t_deceleration.
Substituting ω_f = 0 rad/s and ω_i = final angular velocity from step 2, solve for α_deceleration.
4. Total angle turned:
The total angle turned is the sum of the angles covered during the period of constant angular acceleration (θ_acc) and the period of constant angular deceleration (θ_deceleration).
Total angle = θ_acc + θ_deceleration
Now, using the calculated values, find the total angle turned and the time at which the wheel stops.