A torque of 50 N*m acts on a wheel of moment of inertia 25 kg*m^2 for 4 s and then is removed.
b)How many revolutions does it make in 15 s if it starts at rest?
I found the angular acceleration, 2 rad/s^2, but I don't know how to find the amount of revolutions.
it accelerates for 4 s
... up to 8 rad/s
then spins for 11 s
... 88 rad
the ave vel during acceleration is
... ) + 8) / 2 = 4 rad/s
... spins 16 rad
104 rad total
... divide by 2π to find revolutions
To find the number of revolutions made by the wheel in 15 seconds, we need to first determine the final angular velocity (ωf) of the wheel after 4 seconds of torque application.
Given:
Torque (τ) = 50 N*m
Moment of inertia (I) = 25 kg*m^2
Time (t) = 4 s
To find angular acceleration (α), we can use the equation:
τ = I * α
Rearranging the equation gives:
α = τ / I
Substituting the given values, we have:
α = 50 N*m / 25 kg*m^2
= 2 rad/s^2
Now, we can find the final angular velocity (ωf) using the equation:
ωf = ωi + α * t
Since the wheel starts at rest (ωi = 0), the equation becomes:
ωf = α * t
= 2 rad/s^2 * 4 s
= 8 rad/s
Next, we can find the number of revolutions using the relationship:
Number of revolutions = ωf * t / 2π
Substituting the given values, we have:
Number of revolutions = 8 rad/s * 15 s / (2π)
= (8 * 15) / (2π)
= 120 / (2π)
≈ 19.09 revolutions
Therefore, the wheel will make approximately 19.09 revolutions in 15 seconds starting from rest.
To find the number of revolutions the wheel makes in 15 seconds, we can use the equations of rotational motion.
We have the initial angular velocity, final angular velocity, angular acceleration, and time. We can use the equation:
θ = ω_i * t + 0.5 * α * t^2
where:
θ is the angle (in radians) covered by the wheel,
ω_i is the initial angular velocity,
t is the time,
α is the angular acceleration.
We know that the initial angular velocity is 0, as the wheel starts at rest.
We found the angular acceleration to be 2 rad/s^2.
Substituting these values, we can re-write the equation as:
θ = 0.5 * α * t^2
Plugging in the values:
θ = 0.5 * 2 rad/s^2 * (15 s)^2
= 0.5 * 2 rad/s^2 * 225 s^2
= 225 rad
To convert radians to revolutions, we use the conversion factor:
1 revolution = 2π radians
θ_rev = θ / (2π)
θ_rev = 225 rad / (2π)
= 35.76 revolutions (approximately)
Therefore, the wheel makes approximately 35.76 revolutions in 15 seconds if it starts at rest.