Athin uniform cylindrical turntable of radius 3.3 m and mass 28 kg rotates in a horizontal plane with an initial angular speed of 8.5 rad/s. the turntable bearing is frictionless. a clump of clay of mass 10 kg is dropped onto the turntable and sticks at a point 1.8 m from the point of rotation. find the angular speed of the clay and turntable. calculate the loss of energy to thermal energy.

The angular speed is 7.01 rad/s

The loss of energy to thermal energy is 965.6 J

Explanation:

Conservation of angular momentum says that 'the angular momentum must be the same before and after the clump of clay is attached to the turntable'.

The angular momentum is defined as: L = I*ω

with I = the moment of inertia

with w= the angular speed

The angular momentum before must be equal to the angular momentum after.

Angular momentum before: Lb = It*ω0

Angular momentum after: La = (It + Ic)*ω1

So Lb = La⇔ It*ω0  = (It + Ic)*ω1

with ω1 = the angular speed after (ω1 = ω0*[It/(It + Ic)])

ω₁ = I₀ω₀ / I₁

I₀ = The moment of inertia of the turntable

= The moment of inertia of a cylinder with mass m and radius r around its z axis = ½mr²., and can be calculated as followed:

I₀ = ½(28)(3.3)² = 152.46 kgm²

I₁ = I₀ + I(clay) = 152.46kgm² + m(clay)*r(clay)² = 152.46 kgm² + (10)(1.80)² = 184.86 kgm²

To calculate ω₁ we use the following formula:

ω₁ = I₀ω₀ / I₁ = 152.46*8.5/ 184.86 = 7.01 rad/s

Calculate the loss of energy to thermal energy

Energy is not conserved. Some amount of energy is lost when the clay deforms and sticks to the turntable! This change in energy is given by:

Et= E₁ - E₀  = (0.5)(184.86)(11.078)² - (0.5)(61.8)(4π)² = -577.94 J

with  E₁= ½I₁ω₁² = ½ * 184.86 * 7.01² = 4542.02

with E₀ = ½I₀ω₀² =  ½ * 152.46 *8.5² = 5507.62

Et= E₁ - E₀ = 4542.02 - 5507.62 = - 965.6 J

This value is negative, this means there is 965.6 J energy lost to thermal energy