A wheel on a game show is given an initial angular speed of 1.33 rad/s. It comes to rest after rotating through 3/4 of a turn.
(a) Find the average torque exerted on the wheel given that it is a disk of radius 0.76 m and mass 6.4 kg.
? N·m
sorry, I'm having converting issues...I don't see the link between theta, omega, moment of inirtia, torque...etc...
Torque=momentInertia*deacceleration
The average speed during deacceleration is 1.33/2, so the time to stop is displacement/avg velocity,or .75*2PI/(1.33/2)
deacceleration = wi/time
so you can find torque.
To find the average torque exerted on the wheel, we can use the relationship between torque and angular acceleration.
The moment of inertia (I) represents how the mass of an object is distributed around its axis of rotation. For a disk, the moment of inertia is given by the equation:
I = (1/2) * m * r^2
where m is the mass of the disk and r is the radius. Let's substitute the given values into this equation:
I = (1/2) * 6.4 kg * (0.76 m)^2 = 1.463 kg·m^2
The angular acceleration (α) is the rate at which angular velocity changes. It can be determined using the equation:
α = (ωf - ωi) / t
where ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time taken. The final angular velocity can be calculated by converting 3/4 of a turn into radians:
θ = (3/4) * 2π radians = (3/4) * 6.28 = 4.71 radians
Since the wheel comes to rest, the final angular velocity (ωf) is 0 rad/s. We're given the initial angular velocity (ωi) as 1.33 rad/s. Let's calculate the time (t) taken using the formula:
t = θ / ωi
t = 4.71 radians / 1.33 rad/s = 3.54 seconds
Now, we can substitute the values of ωf, ωi, and t into the equation for angular acceleration to find α:
α = (0 - 1.33 rad/s) / 3.54 s = -0.376 rad/s^2
Finally, the torque (τ) exerted on the wheel can be calculated using the equation:
τ = I * α
τ = 1.463 kg·m^2 * -0.376 rad/s^2 = -0.550 N·m
The average torque exerted on the wheel is -0.550 N·m. The negative sign indicates that the torque is in the opposite direction to the initial angular velocity.