Chapter 9.2, Problem 52E

To determine

To find: The force exert by the rope on the cart to keep the cart rolling down the ramp.

Answer to Problem 52E

The force exerted by the rope to keep the cart rolling down is 21 lb.

Explanation of Solution

Given:

The weight of the cart is 40 lb.

The angle inclined by the ramp to the horizontal is $15°$.

The angle inclined by the rope to the horizontal is $60°$.

Calculation:

The figure below shows that the cart exerts a force of 40 lb in downward direction.

Figure (1)

From Figure (1), it can be observed that the car exerts force into two components u and v. The angle of inclination is $\theta$.

The angle formed between u and w is,

$\begin{array}{c}\theta =90°-15°\\ =75°\end{array}$

The magnitude of the force exerted by the cart on the ramp at an angle of $75°$ is,

$\left|u\right|=w\cos 75°$

The magnitude of the force that holds the cart to roll down ramp at an angle of $75°$ is,

$\left|v\right|=w\cos 75°$

The amount of force applied by the rope to which cart is tied is equal to the amount force that hold the cart to roll down from the ramp.

The force exerted by the rope on the cart to keep it rolling down the ramp is,

$F\cos \theta =w\cos 75°$

Substitute 40 for w and $60°$ for $\theta$ in the above formula.

$\begin{array}{c}F\cos 60°=w\cos 75°\\ F\times 0.5=40\times 0.26\\ F=\frac{40\times 0.26}{0.5}\\ F=20.8\approx 21\end{array}$

Thus, the force exerted by the rope to keep the cart rolling down is 21 lb.