Sorry new here trying to figure out angular problem with a disc, only have 2 attempts remaining please help me out :D.
When the play button is pressed, a CD accelerates uniformly from rest to 470rev/min in 3.0 revolutions.
If the CD has a radius of 6.0 cm and a mass of 15 g , what is the torque exerted on it?
a Disk? I= 1/2 mr^2
Torque= I*alpha
but alpha= (wf-wi)/time
and time is angular displacement/velocityavg
time= 2PI*3/(1/2 479*2PI/60)
then alpha= 470*2PI/60 / time
do that, then calculate
torque= 1/2 mr^2 * alpha
keep units in mks.
First person who ever explained something to me that detailed thank you!
To find the torque exerted on the CD, we can use the rotational analog of Newton's second law, which states that the net torque acting on an object is equal to the moment of inertia of the object multiplied by its angular acceleration.
Step 1: Calculate the final angular velocity:
Given that the CD accelerates uniformly from rest to 470 rev/min in 3.0 revolutions, we first need to convert the final angular velocity to radians per second.
1 revolution = 2π radians
So, the final angular velocity in radians per minute is:
ω = (470 rev/min) x (2π rad/rev) = 940π rad/min.
To convert this to radians per second, divide it by 60:
ω = (940π rad/min) / 60 = 940π/60 rad/s ≈ 49.36 rad/s.
Step 2: Calculate the angular acceleration:
The angular acceleration can be determined using the formula:
α = Δω / Δt,
where Δω is the change in angular velocity and Δt is the time taken.
Since the initial angular velocity is zero, the change in angular velocity is just the final angular velocity:
Δω = ω = 49.36 rad/s.
Given that it takes 3.0 revolutions for the CD to reach this final angular velocity, we can calculate the time taken:
Time = (number of revolutions)/(angular velocity) = (3.0 rev)/(49.36 rad/s) ≈ 0.061 s.
Now, we can calculate the angular acceleration:
α = Δω / Δt = 49.36 rad/s / 0.061 s ≈ 807.21 rad/s².
Step 3: Calculate the moment of inertia:
The moment of inertia of a disc can be calculated using the formula:
I = (1/2) * m * r^2,
where m is the mass of the CD and r is its radius.
Given that the mass of the CD is 15 g (which we need to convert to kilograms) and the radius is 6.0 cm (which we need to convert to meters):
m = 15 g = 0.015 kg (since 1 kg = 1000 g).
r = 6.0 cm = 0.06 m.
Now, we can calculate the moment of inertia:
I = (1/2) * m * r^2 = (1/2) * 0.015 kg * (0.06 m)^2 ≈ 5.4 x 10^-5 kg * m².
Step 4: Calculate the torque:
Using the formula for torque, where τ is the torque, I is the moment of inertia, and α is the angular acceleration:
τ = I * α.
Substituting the values we calculated:
τ = (5.4 x 10^-5 kg * m²) * (807.21 rad/s²) ≈ 0.0437 Nm.
Therefore, the torque exerted on the CD is approximately 0.0437 Nm.