A torque of 60 N*m acts on a wheel of moment of inertia 30 kg*m^2 for 5 s and then is removed.
a) What is the angular acceleration of the wheel?
b) How many revolutions does it make in 15 s if it starts at rest?
a) The angular acceleration of the wheel is 4 rad/s^2.
b) The wheel will make 6 revolutions in 15 s.
a) To find the angular acceleration of the wheel, we can use the equation:
τ = I * α,
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
Given:
τ = 60 N*m,
I = 30 kg*m^2.
Rearranging the equation, we have:
α = τ / I.
Substituting the given values, we get:
α = 60 N*m / 30 kg*m^2.
Simplifying the units, we have:
α = 2 rad/s^2.
Therefore, the angular acceleration of the wheel is 2 rad/s^2.
b) To find the number of revolutions the wheel makes in 15 seconds, we can use the equation:
θ = ω_i * t + 0.5 * α * t^2,
where θ is the angle traveled, ω_i is the initial angular velocity, α is the angular acceleration, and t is the time.
Given:
ω_i = 0 (since the wheel starts at rest),
t = 15 s.
Substituting the given values and the previously calculated angular acceleration, we get:
θ = 0 * 15 + 0.5 * 2 * 15^2.
Simplifying the equation, we have:
θ = 0 + 0.5 * 2 * 225.
θ = 225.
Therefore, the wheel makes 225 revolutions in 15 seconds if it starts at rest.
To solve this problem, we can use the equations of rotational motion.
a) The equation that relates torque (τ), moment of inertia (I), and angular acceleration (α) is τ = I * α. Rearranging the equation, we have α = τ / I.
We are given the torque (τ) as 60 N*m and the moment of inertia (I) as 30 kg*m^2. Plug in the values into the equation, α = 60 N*m / 30 kg*m^2, and we get α = 2 rad/s^2. Therefore, the angular acceleration of the wheel is 2 rad/s^2.
b) The equation that relates angular displacement (θ), initial angular velocity (ω), angular acceleration (α), and time (t) is θ = ω*t + (1/2)*α*t^2.
Since the wheel starts at rest, the initial angular velocity (ω) is 0. We are given the time (t) as 15 s and the angular acceleration (α) as 2 rad/s^2. Plug in the values into the equation, θ = 0*t + (1/2)*(2 rad/s^2)*(15 s)^2, and simplify to get θ = 1,350 rad.
To convert the angular displacement to the number of revolutions, we need to know that 1 revolution is equal to 2π radians. So, the number of revolutions is θ / (2π) = 1,350 rad / (2π) ≈ 214.56 revolutions.
Therefore, the wheel makes approximately 214.56 revolutions in 15 seconds if it starts at rest.