Chapter 17, Problem 17.135RP

A uniform disk, initially at rest and of constant thickness, is placed in contact with the belt shown, which moves at a constant speed $v=80$ ft/s. Knowing that the coefficient of kinetic friction between the disk and the belt is 0.15, determine (a) the number of revolutions executed by the disk before it reaches a constant angular velocity, (b) the time required for the disk to reach that constant angular velocity.

  

To determine

(a)

The number of revolutions executed by the disk before it reaches a constant angular velocity.

Answer to Problem 17.135RP

Disk will take $106.7$ revolutions before it reaches to constant angular velocity.

Explanation of Solution

Given:

Velocity of disk $V=80ft/\sec ,{\mu }_K=0.15$

Concept used:

Principle of work and energy.

Calculation:

Angular velocity, $\omega =\frac{v}{r}$

Moment of inertia $=I=\frac{1}{2}m{r}^2$

Kinetic Friction $f_f={\mu }_{K}N=0.15N$

Summation of forces at y-direction,

$\begin{array}{l}\sum f_y=N\cos {25}^0-f_f\sin {25}^0-m_g=0\\ m_g=(\cos {25}^0-{\mu }_K\sin {25}^0)N\\ f_f=0.15\times 0.1863mg\\ =0.1779mg.\end{array}$

Work was done

$=W=f_{f}r\theta =0.1779mgr\theta$

Applying the principle of work and energy,

$\begin{array}{l}W=T_2-T_1\\ T_1=0\\ T_2=\frac{I{\omega }_2{}^2}{2}=\frac{m{v}^2}{4}\\ 0.1779mgr\theta =\frac{m{v}^2}{4}\\ \theta =\frac{{80}^2}{4\times 0.1779\times 32.2\times (\frac{5}{12})}\\ \theta =670.34radians\\ =106.7revolutions\end{array}$

Conclusion:

We can calculate the number of rotations done by disk before reaching constant angular velocity are $106.7$ rotations.

To determine

(b)

The time taken by disk to reach the constant angular velocity.

Answer to Problem 17.135RP

The time required for the disk to reach the constant angular velocity is $6.98sec.$

Explanation of Solution

Given:

Velocity of disk $V=80ft/\sec ,{\mu }_K=0.15$

Concept used:

Principle of impulse and momentum.

Calculation:

According to impulse momentum principle.

Taking moments about A,

$\begin{array}{l}{f}_f{t}_r=I{\omega }_2\\ t=\frac{I{\omega }_2}{f_{f}r}\\ =\frac{m{r}^2\times v}{2\times 0.1779m\times g\times r^2}\\ =\frac{80}{2\times 0.1779\times 32.2}\\ t=6.98\end{array}$

Conclusion:

Thus, the time taken by disk to reach a constant angular velocity is $6.98sec$ by using the impulse-momentum principle.