Question

Question:51. A truck is moving with constant acceleration aup a hill that makes an angle $\mathit{\varphi }$with the horizontal as in Figure P6.51. A small sphere of mass mis suspended from the ceiling of the truck by a light cord. If Figure P6.51 the pendulum makes a constant angle θwith the perpendicular to the ceiling, what is a?

Short answer

The expression for the acceleration is $a=g(\cos \varphi \tan \theta -\sin \varphi )$.

Step-by-step solution

Expression for Newton’s second law

The expression for the force using Newton’s second law is given by

$F=ma$

HereF is the force,m is the mass of the body anda is the acceleration.

Thus, the magnitude of force is equal to the product of mass and acceleration.

Drawing the free body diagram

The free body diagram of the system is shown below.

Finding the expression for the acceleration

Consider the expression for the Newton’s second law inx direction.

$\begin{array}{c}\sum F_x=T\sin \theta -mg\sin \varphi \\ =ma\end{array}$...... (1)

Consider the expression for the Newton’s second law inx direction.

$\begin{array}{c}\sum F_y=T\cos \theta -mg\cos \varphi \\ =0\end{array}$

Solve for T.

$T=\frac{mg\cos \varphi }{\cos \theta }$...... (2)

Substitute the value of Tfrom equation (2) into equation (1).

$\frac{mg\cos \varphi }{\cos \theta }\sin \theta -mg\sin \varphi =maa$

Solve for a.

$\begin{array}{l}a=g\cos \varphi \tan \theta -g\sin \varphi \\ a=g(\cos \varphi \tan \theta -\sin \varphi )\end{array}$

Therefore, the expression for the acceleration is $a=g(\cos \varphi \tan \theta -\sin \varphi )$.