Subscribe $4.99/month

Un-lock Verified Step-by-Step Experts Answers.

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

Our Website is free to use.
To help us grow, you can support our team with a Small Tip.

All the data tables that you may search for.

Need Help? We got you covered.

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

Products

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

Need Help? We got you covered.

Chapter 20

Q. 20.5

A wheel of mass M has been strengthened using a metal ring. It consists of a uniform disc of radius r and mass σ per unit area surrounded by a uniform solid ring (the rim) of radius r, negligible width, and mass 5rσ per unit length.
i) Find the mass of the two parts of the wheel in terms of M.
ii) Write down the moment of inertia of each part about the axle of the wheel.
iii) Find the kinetic energy when the wheel is rotating with an angular speed ω.
iv) Write this kinetic energy as a percentage of the kinetic energy of a hoop with the same mass and radius rotating at the same angular speed.

Step-by-Step

Verified Solution

i)    The area of the disc is πr² , so its mass is M_{1} = σπr².

The length of the ring is 2πr, so its mass is

M_{2} = 2πr(5rσ)

= 10 σπr² .

Hence                         M = 11 σπr²

⇒         M_{1} = \frac{M}{11}  and  M_{2} = \frac{10M}{11}.

ii) The moment of inertia of the inside disc about the axle is

I_{2} = \frac{1}{2}M_{1}r^{2}

= \frac{Mr^{2}}{22}.

  The moment of inertia of the rim about the axle is

I_{2} =M_{2}r^{2}

= \frac{10Mr^{2}}{11}.

iii) The kinetic energy of the wheel is \frac{1}{2}Iω², where I = I_{1}  +  I_{2}.

I_{1}  +  I_{2} = \frac{Mr^{2}}{22}  +  \frac{10Mr^{2}}{11}

= \frac{21}{22}Mr^{2}

iv) The moment of inertia of the hoop about its axis is Mr², so its kinetic energy is \frac{1}{2}Mr^{2}ω^{2}.

The kinetic energy of the wheel is \frac{21}{22} × 100% of the kinetic energy of the hoop, namely 95.5%.

fig 20.12