Question

at the local discotheque the speed dial controlling the 8 kg solid and uniform disco ball of radius 30 cm malfunctioned causing the disco ball to accelerate from 7 revolutions per minute up to 60 revolutions per minute. calculate the change in kinetic energy.

Answer 1

The disco ball increased its kinetic energy by approximately 5.673 Joules as it accelerated from 7 rpm to 60 rpm, calculated using the formulas for rotational kinetic energy and the moment of inertia for a solid sphere

Step-by-step explanation:

We need to calculate the change in kinetic energy of the disco ball as it accelerates from 7 revolutions per minute (rpm) to 60 rpm. The kinetic energy of a rotating object is given by KE = (1/2)Iω2, where I is the moment of inertia and ω is the angular velocity in radians per second.

Firstly, we convert the angular velocities from rpm to radians per second:

• Initial angular velocity: 7 rpm × (2π rad/rev) / (60 s/min) = 0.733 rad/s
• Final angular velocity: 60 rpm × (2π rad/rev) / (60 s/min) = 6.283 rad/s

The moment of inertia for a solid sphere is I = (2/5)mR2, where m is the mass and R is the radius.

Plugging the given values,

I = (2/5)(8 kg)(0.3 m)2 = 0.288 kg·m2

Now we calculate the kinetic energies at the two angular velocities:

• Initial KE: (1/2)(0.288 kg·m2)(0.733 rad/s)2 ≈ 0.077 J
• Final KE: (1/2)(0.288 kg·m2)(6.283 rad/s)2 ≈ 5.75 J

The change in kinetic energy is the final kinetic energy minus the initial kinetic energy:

ΔKE = 5.75 J - 0.077 J ≈ 5.673 J

The change in kinetic energy is approximately 5.673 Joules.