Chapter 6, Problem 91GP

To determine

a.The distance travelled by the sled up the inclined plane

b. The condition on the coefficient of static friction for the sled not to get stuck

c. The speed of the sled when it slides back down

Answer to Problem 91GP

Solution:

a. The distance travelled by the sled up the inclined plane is $0.40\text{ m}$ .

b. The coefficient of static friction must be less than $0.53$ .

c. The speed of the sled when it slides back down is $1.4\text{ m/s}$ .

Explanation of Solution

To understand the question, let us check the following figure.

  

Given:

The mass of the sled is $m=18\text{kg}$ .

The inclination angle is $\theta =28°$

The speed of the moving up is $v=2.3\frac{\text{m}}{\text{s}}$

The coefficient of kinetic friction is ${\mu }_k=0.25$ .

Formula used:

From energy conservation, the kinetic energy of the sled is equal to the friction and gravitational potential energy summation.

  $\frac{1}{2}m{v}^2=mgh+Fd\mathrm{....}(1)$

Here, F is the friction force.

When the sled slides back, the energy conservation equation is,

  $\frac{1}{2}m{v}^2=mgh-Fd\mathrm{....}(2)$

Calculations:

a.

From equation (1)

  $\begin{array}{l}\frac{1}{2}m{v}^2=mgh+Fd\\ \frac{1}{2}m{v}^2=mgd\sin 28°+{\mu }_{k}mgd\cos 28°\\ \frac{1}{2}{v}^2=gd(\sin 28°+{\mu }_k\cos 28°)\\ d=\frac{v^2}{2g(\sin 28°+{\mu }_k\cos 28°)}\\ d=\frac{{2.3}^2}{2\times 9.8\times (0.47+0.25\times 0.88)}\text{m}\\ d=\frac{{2.3}^2}{13.5}\text{m}\\ d=0.40\text{m}\end{array}$

b.

For the sled not to get stuck, the coefficient of static friction has to be less than

  $\begin{array}{l}{\mu }_s<\frac{\sin 28°}{\cos 28°}\\ {\mu }_s<0.53\end{array}$

c.

From equation (2)

  $\begin{array}{l}\frac{1}{2}m{v}^2=mgh-Fd\\ \frac{1}{2}m{v}^2=mgd\sin 28°-{\mu }_{k}mgd\cos 28°\\ \frac{1}{2}{v}^2=gd(\sin 28°-{\mu }_k\cos 28°)\\ v^2=2gd(\sin 28°-{\mu }_k\cos 28°)\\ v^2=2\times 9.8\times 0.40\times (0.47-0.25\times 0.88)\\ v=1.4\frac{\text{m}}{\text{s}}\end{array}$

Conclusion:

a. The distance travelled by the sled up the inclined plane is $0.40\text{ m}$ .

b. The coefficient of static friction must be less than $0.53$ .

c. The speed of the sled when it slides back down is $1.4\text{ m/s}$ .