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 6.8×10−3kg⋅m2 .

KE = 1/2*Iω^2

ω is the rotational velocity, but the unit is in rad/s. rpm means "revolutions per minute"
You need to convert 33 rpm to rad/s first:
33 rev/min = 33/60 rev/s = 0.55 rev/s
1 rev = 2π rad: => 0.55 rev/s*2π rad/rev = 0.55*2π rad/s = 1.1π rad/s ≈ 3.45575 rad/s

Now with the KE formula: KE = 1/2*Iω^2 = (1/2)*(6.0×10^-3)*(1.1π)^2 = 0.035827 J

Well, we can calculate the change in rotational kinetic energy using the formula:

ΔKE = 1/2 * I * (ω2^2 - ω1^2)

Where:
ΔKE is the change in rotational kinetic energy
I is the rotational inertia
ω2 is the final angular velocity
ω1 is the initial angular velocity

First, let's convert the angular velocities from rpm to rad/s. Since 1 revolution is equal to 2π radians, we have:

ω1 = 0 rpm * (2π rad/1 min) * (1 min/60 s) = 0 rad/s
ω2 = 33 rpm * (2π rad/1 min) * (1 min/60 s) = 2.18 rad/s

Now, we can plug these values into the formula:

ΔKE = 1/2 * 6.8×10^-3 kg⋅m^2 * (2.18^2 - 0^2)

Calculating that out:

ΔKE = 1/2 * 6.8×10^-3 kg⋅m^2 * 4.7524

ΔKE ≈ 0.0727 Joules

So, the change in rotational kinetic energy is approximately 0.0727 Joules. But hey, don't worry, the stereo system won't be spinning any jokes with that energy change!

To determine the change in rotational kinetic energy, we can use the formula:

ΔKE = ½ * I * (ω^2 - ω₀^2)

Where:
ΔKE = change in rotational kinetic energy
I = rotational inertia
ω = final angular velocity
ω₀ = initial angular velocity

Given:
I = 6.8 × 10^(-3) kg⋅m^2
ω₀ = 0 rpm (initial angular velocity)
ω = 33 rpm (final angular velocity)

First, we convert the given angular velocities from rpm to rad/s by using the conversion factor of 1 rpm = (2π/60) rad/s:

ω₀ = 0 rpm = 0 rad/s
ω = 33 rpm = (33 * 2π/60) rad/s = 11π/10 rad/s

Now we can substitute the values into the formula and calculate the change in rotational kinetic energy:

ΔKE = ½ * (6.8 × 10^(-3)) * ((11π/10)^2 - (0)^2)
= ½ * (6.8 × 10^(-3)) * ((121π^2/100) - 0)
= (6.8 × 10^(-3)) * (121π^2/200)

Finally, we can simplify the expression:

ΔKE = (6.8 × 121π^2 × 10^(-3)) / 200
= 0.5208π^2 × 10^(-3)

Therefore, the change in rotational kinetic energy when the rotational velocity of the turntable increases from 0 to 33 rpm is approximately 0.5208π^2 × 10^(-3) joules.

To determine the change in rotational kinetic energy, we need to calculate the initial and final rotational kinetic energies, and then find the difference between the two.

First, let's calculate the initial rotational kinetic energy. When the rotational velocity is 0, the initial rotational kinetic energy is also 0 since kinetic energy depends on the square of the velocity.

Next, let's calculate the final rotational kinetic energy. We are given that the rotational velocity increases from 0 to 33 rpm. However, we need to convert this angular velocity to radians per second to use it in the formula.

1 revolution = 2π radians
1 minute = 60 seconds

Therefore, 33 rpm = (33 * 2π) / 60 radians per second.

Now, we can calculate the final rotational kinetic energy using the formula:

Rotational kinetic energy = (1/2) * I * ω^2

where I is the rotational inertia and ω is the angular velocity in radians per second.

Plugging in the values, we have:

Final rotational kinetic energy = (1/2) * (6.8×10−3 kg⋅m^2) * ((33 * 2π) / 60)^2

Calculating this gives us the final rotational kinetic energy.

Finally, to find the change in rotational kinetic energy, we subtract the initial value (which is 0) from the final value. The change in rotational kinetic energy is equal to the final rotational kinetic energy.

So, the change in rotational kinetic energy when the rotational velocity increases from 0 to 33 rpm is equal to the final rotational kinetic energy we calculated earlier.