Q: A solid disc flywheel of mass $$8 \times 10^4$\ \mathrm{kg}$$ and diameter $$5.0\ \mathrm{m}$$ can spin at a max of $$120\ \mathrm{RPM}. $$What is the maximum energy it can store?

$$E = 4.93 \times 10^7$\ \mathrm{J}$$

Question 1

A car tire of radius 0.40 m rotates 10,000 times while driving. How far did the car travel in meters? Use $$\Delta x = r \Delta \Theta$$ and recall that one revolution is $$2\pi$$ radians.

Accepted Answer

$$25,132.74 m$

Question 2

A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. What is its angular acceleration in $$\mathrm{$$rad/s^2$$}$$? (Recall $$1\ \mathrm{RPM} = \frac{2\pi}{60}\ \mathrm{rad/s})$$

Accepted Answer

$$\alpha = 10\pi\ \mathrm{$$rad/s^2$$}$$

Question 3

A solid disc of radius 2 m and mass 10 kg spins at 120 RPM. What is its moment of inertia about its central axis?

Accepted Answer

$$I = 40\ \mathrm{kg\ m^2$}$$

Question 4

A 4 m long blade initially at rest begins to spin with $$3\ \mathrm{$$rad/s^2$$}$$ around its axis at the middle. After 10 s, what is the tangential speed at the tip?

Accepted Answer

$$v = 120\ \mathrm{m/s}$$

Question 5

A solid sphere of mass $$2\ \mathrm{kg}$$ and radius $$0.3\ \mathrm{m}$$ rolls without slipping at $$10\ \mathrm{m/s}. $$What is its total kinetic energy?

Accepted Answer

$$K = 150\ \mathrm{J}$$

Question 6

A composite disc consists of a solid disc (mass 4 kg, radius 2 m) and a thick-walled hoop (mass 5 kg, inner radius 2 m, outer radius 3 m). What is the total moment of inertia about the central axis?

Accepted Answer

$$I = 40.5\ \mathrm{kg\ m^2$}$$

Question 7

A hollow sphere of mass $$M$$ and radius $$R$$ rolls without slipping with angular speed $$\omega. $$What is the ratio of its linear kinetic energy to its total kinetic energy?

Accepted Answer

$$\frac{2}{3}$$

Question 8

A yo-yo (modeled as a solid disc, mass $$0.1\ \mathrm{kg}$$, radius $$0.02\ \mathrm{m}) is $$released from rest and falls, unwinding its string. What is its linear acceleration?

Accepted Answer

$$a = \frac{2}{3}g$$

Question 9

A solid cylinder of mass $$10\ \mathrm{kg}$$ and radius $$2\ \mathrm{m} is $$initially at rest. How much work is needed to accelerate it to $$120\ \mathrm{RPM}$$?

Accepted Answer

$$W = 316.8\ \mathrm{J}$$

Question 10

A block of mass $$m$$ hangs from a rope wrapped around a pulley (solid cylinder, mass $$M$$, radius $$R). $$When released, what is the acceleration of the block?

Accepted Answer

$$a = \frac{m}{m + \frac{1}{2}M}g$$

Question 11

A solid disc flywheel of mass $$8 	imes 10^4$\ \mathrm{kg}$$ and diameter $$5.0\ \mathrm{m}$$ can spin at a max of $$120\ \mathrm{RPM}. $$What is the maximum energy it can store?

Accepted Answer

$$E = 4.93 	imes 10^7$\ \mathrm{J}$$

Question 12

A solid sphere of mass $$M$$ and radius $$R is $$released from rest at the top of an incline of height $$h$$ and rolls without slipping. What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{\frac{10}{7}gh}$$

Question 13

A thin rod of mass $$M$$ and length $$L$$ rotates about an axis at its end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3}ML^2$

Question 14

A bicycle's back sprocket has radius $$r_2$$ and the middle sprocket has radius $$r_1. If $$the pedals spin at angular speed $$\$$omega_1$$, what is the angular speed $$\$$omega_2 of $$the back sprocket (assuming the chain does not slip)?

Accepted Answer

$$\$$omega_2$$ = \$$omega_1$$ \frac{r_1$}{r_2$}$$

Question 15

A solid disc of radius $$R$$ and mass $$M$$ rotates about an axis at its rim. What is its moment of inertia using the parallel axis theorem?

Accepted Answer

$$I = \frac{3}{2}MR^2$

Question 16

A 2-kg object spins in a circle of radius 4 m at $$3\ \mathrm{rad/s}. $$What is its kinetic energy using the rotational formula?

Accepted Answer

$$K = 72\ \mathrm{J}$$

Rotation of Rigid Bodies – Comprehensive Study Notes

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