Chapter 9.5, Problem 31AYU

To determine

To find: How many pounds of force must Timmy exert to hold the piano in position?

Answer to Problem 31AYU

Solution:

$85.5$ lbs.

Explanation of Solution

Given:

Billy and Timmy are using a ramp to load furniture into a truck. While rolling a 250-pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is ${20}^{\circ }$.

Calculation:

Angles of $A$ and $B$ form the alternate angles, so they are equal.

We wish to find the magnitude of the force $v$ that is acting to cause the piano up the ramp. A force with the same magnitude in the opposite direction of $v$ will keep the piano in place on the ramp. The force of gravity is orthogonal to the level ground, so the force of the wagon due to gravity can be represented by the vector,

$F_g=-250j$

Determine the vector projection of $F_g$ onto $w$,

$w=\cos {20}^{\circ }i+\sin {20}^{\circ }j$

The vector projection of $F_g$ onto $w$ is,

$v=\frac{F_g\cdot w}{\Vert w{\Vert }^2}w=-\frac{250\left(\sin {20}^{\circ }\right)}{\sqrt{{\cos }^2{20}^{\circ }+{\sin }^2{20}^{\circ }}}\left(\cos {20}^{\circ }i+\sin {20}^{\circ }j\right)$

$=-85.5\left(\cos {20}^{\circ }i+\sin {20}^{\circ }j\right)$

The magnitude of $v$ is $85.5$ pounds, so the magnitude of the force required to keep the piano up the ramp is $85.5$ pounds.