A solid disk of radius 0.20 m and a mass of 1.0 kg starts at rest and then speeds up to 100 rad/s in 10s. What is the torque that produces this rotational acceleration?
ω=ε•t
ε=ω/t=100/10=10 rad/s²
M=I•ε=(m•R²/2) • ε= 1•0.2²•10/2=0.5 N•m
To find the torque that produces this rotational acceleration, we can use the formula:
Torque (τ) = Moment of Inertia (I) × Angular Acceleration (α)
First, let's find the moment of inertia for a solid disk about its axis of rotation. The moment of inertia for a solid disk can be calculated using the formula:
I = (1/2) × mass × radius^2
Given that the radius (r) of the disk is 0.20 m and the mass (m) of the disk is 1.0 kg, we can substitute these values into the formula to find the moment of inertia (I):
I = (1/2) × 1.0 kg × (0.20 m)^2
= 0.02 kg·m^2
Next, we need to find the angular acceleration (α). The angular acceleration can be calculated using the formula:
α = (final angular velocity - initial angular velocity) / time
Given that the initial angular velocity (ω_i) is 0 rad/s, the final angular velocity (ω_f) is 100 rad/s, and the time (t) is 10 s, we can substitute these values into the formula to find the angular acceleration (α):
α = (100 rad/s - 0 rad/s) / 10 s
= 10 rad/s^2
Finally, we can substitute the moment of inertia (I = 0.02 kg·m^2) and the angular acceleration (α = 10 rad/s^2) into the torque formula to calculate the torque (τ):
τ = I × α
= 0.02 kg·m^2 × 10 rad/s^2
= 0.20 N·m
Therefore, the torque that produces this rotational acceleration is 0.20 N·m.
To find the torque that produces this rotational acceleration, we can use the following formula:
Torque (τ) = Moment of Inertia (I) * Angular Acceleration (α)
The moment of inertia (I) of a solid disk is given as:
I = (1/2) * m * r^2
where m is the mass of the disk and r is the radius. In this case, m = 1.0 kg and r = 0.20 m.
First, let's calculate the moment of inertia:
I = (1/2) * 1.0 kg * (0.20 m)^2
= 0.02 kg·m^2
Next, let's calculate the angular acceleration (α):
Angular acceleration (α) = (Final angular velocity - Initial angular velocity) / Time
Given the initial angular velocity (ω₀) is 0 rad/s, final angular velocity (ω) is 100 rad/s, and time (t) is 10 s, we have:
α = (100 rad/s - 0 rad/s) / 10 s
= 10 rad/s^2
Now, we can substitute the values of moment of inertia (I) and angular acceleration (α) into the torque formula:
Torque (τ) = 0.02 kg·m^2 * 10 rad/s^2
= 0.2 N·m
Therefore, the torque that produces the rotational acceleration is 0.2 N·m (Newton meters).