A Pendulum and a Truck

Classical Mechanics Level 4

A flatbed truck has a box of mass $M$ riding on the bed of the truck. There is a hanger on the box which supports a simple pendulum. As the truck accelerates with acceleration $A$ , the box slides toward the rear of the truck but, due to friction, experiences an acceleration of $a$ in the forward direction. This causes the string of the pendulum to make an angle of $\theta$ with the vertical direction.

What is the measure of angle $\theta$ to the nearest tenth of a degree?

Details and Assumptions:

Assume that the pendulum bob has a negligible mass of $m$ .
The coefficient of kinetic friction between the box and the truck bed is $\mu_{k}=0.5774$ .

The answer is 30.0.

1 solution

Dale Gray
Jan 5, 2017

By making use of the equivalence principle, one can consider the pendulum bob as being supported by a string aligned with an effective gravitational field $g^{\prime}$ which is the vector sum of the Earth's gravitational field, $g$ , and a fictitious gravitational field $-a$ as in the picture below.

One has $\tan\theta =\dfrac{a}{g}$ .

To find $a$ consider that in an inertial frame the box is accelerating due to a net force $f_k = \mu_k N = Ma$ , where $N$ is the normal force of the truck bed on the box. Using $N = Mg$ gives

$Ma = \mu_k Mg .$

Therefore,
$\tan \theta = \dfrac ag = \mu_k = 0.5774$ .
Thus $\theta = 30.0^\circ$ .

I'm happy to support you.

Jongheun Lee Staff - 4 years, 5 months ago

Thanks again, to whomever, for making improvements in my drawing!

Dale Gray - 4 years, 5 months ago

A Pendulum and a Truck

The answer is 30.0.

1 solution

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