A torque of 77.7 Nm causes a wheel to start from rest , completes 5.55 revolutions and attains a final angular velocity of 88.8 rad/sec. What is the moment of inertia of the wheel?
To find the moment of inertia of the wheel, we can use the following formula:
Torque = Moment of Inertia x Angular Acceleration
Given that the torque is 77.7 Nm, angular velocity is 88.8 rad/sec, and 5.55 revolutions are completed, we can calculate the angular acceleration:
Initial angular velocity = 0 rad/sec
Final angular velocity = 88.8 rad/sec
Number of revolutions = 5.55
Convert revolutions to radians:
5.55 revolutions x 2π radians/revolution = 11π radians
Using the equation for angular acceleration:
Angular acceleration = (Final angular velocity - Initial angular velocity) / Time
Angular acceleration = (88.8 rad/sec - 0 rad/sec) / Time
Since the wheel starts from rest, the time taken for the wheel to complete the 5.55 revolutions can be calculated using the equation for the number of revolutions:
2π x (5.55) = 11π
θ = ω_initial x t + (1/2) x angular acceleration x t^2
11π = 0 x t + (1/2) x angular acceleration x t^2
11π = (1/2) x angular acceleration x t^2
Therefore, angular acceleration = (2 x 11π) / t^2
Angular acceleration = (22π) / t^2
Plugging in the values:
(22π) / t^2 = (88.8) / t
Solving for t:
t = (22π) / 88.8
t = 2π
Now that we have found the time taken for the wheel to complete the revolutions, we can calculate the angular acceleration:
Angular acceleration = (88.8 rad/sec - 0 rad/sec) / (2π)
Angular acceleration = 88.8 rad/sec / 2π
Angular acceleration = 14.1 rad/sec^2
Now, to find the moment of inertia:
77.7 = Moment of Inertia x 14.1
Moment of Inertia = 77.7 / 14.1
Moment of Inertia = 5.5 kg.m^2
Therefore, the moment of inertia of the wheel is 5.5 kg.m^2.