Question

A small sphere is hung by a string from the ceiling of a van. When the van is stationary, the sphere hangs vertically. However, when the van accelerates, the sphere swings backward so that the string makes an angle of θ with respect to the vertical. Find the acceleration of the van when θ = 13.2°.

Answer 1

A small sphere is hung by a string from the ceiling of a van. When the van is stationary, the sphere hangs vertically. However, when the van accelerates, the sphere swings backward so that the string makes an angle of $\theta$ with respect to the vertical. Find the acceleration of the van when $\theta = 13.2{}^{\circ}$.

Solution:

a – acceleration of the van;

$\theta = 13.2{}^{\circ}$ – angle that the string makes with respect to the vertical;

m – mass of the sphere

When the van accelerates, the ball relative to the van is in a non-inertial reference frame, because $V \neq$ const. Because of this on the sphere begins to act inertial force directed against the motion of the van:

$$F_{i} = ma$$

Right triangle ABC:

$$\tan \theta = \frac{F_{i}}{mg}$$

(1) in (2):

$$\tan \theta = \frac{ma}{mg}$$ $$a = g \cdot \tan \theta = 9.8 \frac{m}{s^{2}} \cdot \tan 13.2{}^{\circ} = 2.3 \frac{m}{s^{2}}$$

Answer: acceleration of the van is $2.3 \frac{m}{s^{2}}$