Determine the change in rotational kinetic energy when the rotational velocity of the turntable of a stereo system increases from 0 to 33 rpm. Its rotational inertia is 6.2×10−3kg⋅
Rotational Kinetic Energy = Work done by torque = (1/2)*(Moment of inertia)*(Final angular Velocity)^2
Convert rotations per minute to radians per second, plug in that value along with the value of the Moment of inertia (rotational inertia) and solve for kinetic energy.
To determine the change in rotational kinetic energy, we need to know the formula for rotational kinetic energy and apply it to the given information.
The formula for rotational kinetic energy is given by:
K = (1/2) I ω^2
where:
K is the rotational kinetic energy,
I is the rotational inertia, and
ω is the angular velocity.
In this case, we are given the rotational inertia as 6.2×10−3 kg⋅m^2 and the change in angular velocity as 33 rpm. However, we need to convert the angular velocity from rpm (revolutions per minute) to radians per second (rad/s) since the formula requires the angular velocity in rad/s.
To convert from rpm to rad/s, we use the conversion factor:
1 rpm = (2π/60) rad/s
So, the change in angular velocity is:
33 rpm * (2π/60) rad/s = 33/60 * 2π rad/s = (33/30) π rad/s
Now, we can substitute the values into the formula to find the change in rotational kinetic energy:
ΔK = (1/2) I Δω^2
where ΔK is the change in rotational kinetic energy and Δω is the change in angular velocity.
Plugging in the values:
ΔK = (1/2) * 6.2×10−3 kg⋅m^2 * [(33/30) π rad/s]^2
Simplifying the expression and calculating:
ΔK = (1/2) * 6.2×10−3 * (33/30)^2 * π^2
Now, multiply the numbers together:
ΔK ≈ 0.0169 kg⋅m^2 * π^2
Calculating:
ΔK ≈ 0.0169 * 9.87 ≈ 0.1668 J
Therefore, the change in rotational kinetic energy when the rotational velocity of the turntable increases from 0 to 33 rpm is approximately 0.1668 Joules.