Question

Review: A uniform disk of mass 10.0 kg and radius 0.250 m spins at 300 rev/min on a low-friction axle. It must be brought to a stop in 1.00 min by a brake pad that makes contact with the disk at an average distance 0.220 m from the axis. The coefficient of friction between pad and disk is 0.500. A piston in a cylinder of diameter 5.0 cm presses the brake pad against the disk. Find the pressure required for the brake fluid in the cylinder.

Short answer

The pressure required for the brake fluid in the cylinder is P=758 Pa .

Step-by-step solution

Angular acceleration:

Angular acceleration is defined as the rate of change of angular velocity over time. It is usually expressed in radians per second per second.

The angular acceleration is:

$\alpha =\frac{\Delta \omega }{\Delta t}$

Where, $a$is the angular acceleration, $∆\omega$is the change in angular frequency, $∆t$is the change in time, and

The disk (mass M=10.0kg , radius R=0.250 m) has a moment of inertia is , $l=\frac{1}{2}M{R}^2$

The disk slows from ${\omega }_i=300rev/minto{\omega }_f=0intimeinterval∆t=60.0s$

Its angular acceleration is,

$$\begin{gathered} a=\frac{∆\omega }{∆t} \\ =\frac{{\omega }_f-{\omega }_i}{∆t} \\ =\frac{-{\omega }_i}{∆t} \end{gathered}$$

Frictional torque from the brake pad slows the wheel. Friction has moment arm d=0.220 m

The relation between friction and angular acceleration is:

$$\begin{gathered} f={\mu }_{k}n \\ n=\frac{f}{{\mu }_k} \\ =\frac{M{R}^2{\omega }_i}{2{\mu }_{k}d\Delta t} \end{gathered}$$

The normal force and coefficient of friction $({\mu }_k=0.500)$ between the brake pad and the disk determine the amount of friction. We can write an expression for the normal force:

$$\begin{gathered} P=\frac{n}{\pi r^2} \\ =\frac{M{R}^2{\omega }_i}{2{\mu }_{k}d\pi r^2\Delta t} \\ =\frac{(10.0kg){(0.250m)}^2\left[\frac{300rev\times 2\mathrm{\pi }\mathrm{rad}\times 1\min }{min\times 1rev\times 60.0s}\right]}{2(0.500)(0.220m)(60.0s)\mathrm{\pi }{(0.0250\mathrm{m})}^2} \\ =758Pa \end{gathered}$$

Hence, the pressure required for the brake fluid in the cylinder is P=758 Pa