Question

A 200 g, 40-cm-diameter turntable rotates on frictionless bearings at 60 rpm. A 20 g block sits at the center of the turntable. A compressed spring shoots the block radially outward along a frictionless groove in the surface of the turntable. What is the turntable’s rotation angular velocity when the block reaches the outer edge?

Short answer

The turntable’s angular velocity is 5.76 rad / sec

Step-by-step solution

Given information

mass of turn table = 200 gm = 0.2 kg

Diameter of turntable = 40 cm = 0.4 m

so radius of turn table =0.2 m

mass of block = 20 gm =0.02 kg

rpm = 60.

covert rpm into rad/sec = 2π rad/sec

Explanation

Use the law of conservation for momentum

initial momentum = final momentum

L_i= L_f..............................................(1)

Get initial momentum

L_i= I_iω_i.............................................(2)

Find moment of inertia

$I_i=\frac{1}{2}(M+m)r^2$

Substitute this in equation(2)

$L_i=\left(\frac{1}{2}(M+m)r^2\right){\omega }_i........................\left(3\right)$

Substitute the given value

$$\begin{gathered} L_i=\left(\frac{1}{2}\times (0.2\mathrm{kg}+0.02\mathrm{kg})(0.2\mathrm{m}{)}^2\right)\times 2\pi {\mathrm{rads}}^{-1} \\ =0.0044{\mathrm{kgm}}^2\times 2\pi {\mathrm{rads}}^{-1}\dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(4\right) \end{gathered}$$

Now find the final momentum

$$\begin{gathered} L_f=I_f{\omega }_f \\ \left(I_t+I_{block}\right){\omega }_f \\ =\left(\frac{1}{2}M{r}^2+m{r}^2\right){\omega }_f\dots \dots \dots \dots \dots \dots \dots \dots \dots \left(5\right) \end{gathered}$$

Substitute the given values

$$\begin{gathered} L_f=\left(\frac{1}{2}\times 0.2\mathrm{kg}\times (0.20\mathrm{m}{)}^2+0.02\mathrm{kg}\times (0.2\mathrm{m}{)}^2\right){\omega }_f \\ =\left(0.0048{\mathrm{kgm}}^2\right){\omega }_f...........................\left(6\right) \end{gathered}$$

Equate (4) and (6) we get

${\omega }_f=\frac{\left(0.0044{\mathrm{kgm}}^2\right)\times 2\pi {\mathrm{rads}}^{-1}}{\left(0.0048{\mathrm{kgm}}^2\right)}=5.76{\mathrm{rads}}^{-1}$