When the play button is pressed, a CD accelerates uniformly from rest to 420 rev/min in 5.0 revolutions.
Part A
If the CD has a radius of 6.5 cm and a mass of 20 g , what is the torque exerted on it?
Express your answer using two significant figures. N.m
Time required to get up to speed:
theta = (1/2)(w0 + w)*t
5 rev = (1/2)(0 rev/s + 420/60 rev/s)*t
t = 1/7 s
Angular speed that the CD gets up to:
420/60 rev/sec *2 π rad/rev
= 14 π rad/s
Angular acceleration of CD:
alpha = (14 π rad/s - 0 rad/s)/(1/7)s
alpha = 98 π rad/s^2
Moment of inertia of CD:
I = mk^2
I = (0.020 kg)(0.065 m)^2
I = 8.45 * 10^-5 kg*m^2
Torque exerted on CD:
Ta = I*alpha
Ta = (______)(______)
Ta = ______ N*m
The previous response is correct except that Inertia for a disk or solid cylinder = (1/2)(M)(R^2)
To find the torque exerted on the CD, we need to consider the rotational motion and use the formula for torque:
Torque (τ) = Moment of Inertia (I) * Angular Acceleration (α)
The moment of inertia (I) for a solid disc can be given by the equation:
I = (1/2) * mass * radius^2
Given:
Mass (m) = 20 g = 0.02 kg
Radius (r) = 6.5 cm = 0.065 m
We first need to find the angular acceleration (α) of the CD.
Using the formula for angular acceleration:
Angular acceleration (α) = Change in angular velocity (Δω) / Time (t)
The change in angular velocity is given by:
Δω = Final angular velocity (ωf) - Initial angular velocity (ωi)
Given:
Final angular velocity (ωf) = 420 rev/min
Initial angular velocity (ωi) = 0 rev/min
Converting the angular velocities to radians per second:
ωf = (420 rev/min) * (2π rad/rev) * (1 min/60 s)
ωi = (0 rev/min) * (2π rad/rev) * (1 min/60 s)
Using the formula for change in angular velocity:
Δω = ωf - ωi
Now, we can calculate the torque:
τ = (1/2) * mass * radius^2 * α
Let's go step by step.
Step 1: Convert the given mass to kilograms.
Mass (m) = 20 g = 0.02 kg
Step 2: Convert the given radius to meters.
Radius (r) = 6.5 cm = 0.065 m
Step 3: Convert the angular velocities to radians per second.
ωf = (420 rev/min) * (2π rad/rev) * (1 min/60 s)
ωi = (0 rev/min) * (2π rad/rev) * (1 min/60 s)
Step 4: Calculate the change in angular velocity.
Δω = ωf - ωi
Step 5: Calculate the moment of inertia.
I = (1/2) * mass * radius^2
Step 6: Calculate the angular acceleration.
α = Δω / t
Step 7: Calculate the torque.
τ = I * α
Let's plug in the values and calculate the torque.
To find the torque exerted on the CD, we need to use the rotational analog of Newton's second law, which states that the torque (τ) is equal to the moment of inertia (I) multiplied by the angular acceleration (α). The moment of inertia of a solid disk rotating about its central axis is given by the equation:
I = (1/2) * m * r^2,
where m is the mass of the disk and r is its radius.
First, let's convert the radius from centimeters to meters:
r = 6.5 cm = 0.065 m.
Next, let's convert the mass from grams to kilograms:
m = 20 g = 0.02 kg.
Now, let's calculate the moment of inertia (I):
I = (1/2) * m * r^2
= (1/2) * 0.02 kg * (0.065 m)^2
≈ 0.000042 kg·m^2 (rounding to two significant figures).
Next, we need to find the angular acceleration (α). The angular acceleration can be calculated using the angular velocity (ω) and the time (t) it takes to go from rest to 420 rev/min.
Since the CD is accelerating uniformly, we can calculate the angular acceleration using the formula:
α = (ωf - ωi) / t,
where ωf is the final angular velocity, ωi is the initial angular velocity (which is 0 since the CD starts from rest), and t is the time.
Given that the CD goes from rest to 420 rev/min (or 420/60 = 7 rev/s) in 5.0 revolutions, we have:
ωf = 7 rev/s.
Let's convert this angular velocity to radians per second:
ωf = 7 rev/s * (2π rad/rev)
= 7 * 2π rad/s
≈ 43.99 rad/s (rounding to two significant figures).
Now, we can calculate the angular acceleration (α):
α = (ωf - ωi) / t
= (43.99 rad/s - 0 rad/s) / 5.0 revolutions
= 43.99 rad/s / 5.0 revolutions
≈ 8.798 rad/s^2 (rounding to three significant figures).
Finally, we can find the torque (τ) by multiplying the moment of inertia (I) by the angular acceleration (α):
τ = I * α
= 0.000042 kg·m^2 * 8.798 rad/s^2
≈ 0.00037 N·m (rounding to two significant figures).
Therefore, the torque exerted on the CD is approximately 0.00037 N·m.