Problem — Pendulum on an accelerating vehicle

Problem

A pendulum ($2\;\text{kg}$) hangs from the ceiling of a bus accelerating at $a = 3\;\text{m/s}^2$. Find the magnitude of the tension and the angle from the vertical.

Solution

The pendulum, in relative equilibrium inside the bus, tilts by an angle $\theta$ from the vertical and accelerates horizontally together with the vehicle. Acting on it are the weight $\vv{P}$ (vertical, downward) and the string tension $\vv{t}$ (along the string, tilted by $\theta$). We choose the $x$ axis in the direction of the acceleration and the $y$ axis upward. Newton’s second law in components:

$\begin{cases} t\sin\theta = m a \\ t\cos\theta - m g = 0 \end{cases}$

Angle. Dividing the first equation by the second eliminates $t$:

$\tan\theta = \frac{a}{g} = \frac{3}{9{,}81} = 0{,}306 \quad\Rightarrow\quad \theta = \arctan 0{,}306$

$\ev{\theta \approx 17^\circ}$

Magnitude of the tension. From the second equation $t\cos\theta = mg$ and from the first $t\sin\theta = ma$; squaring and adding ($\sin^2\theta + \cos^2\theta = 1$):

$t = m\sqrt{g^2 + a^2} = 2\sqrt{9{,}81^2 + 3^2} = 2\sqrt{105{,}4}$

$\ev{|\vv{t}| \approx 20{,}5\;\text{N}}$

The string holds more than the simple weight ($mg \approx 19{,}6\;\text{N}$): it must also provide the horizontal component that accelerates the mass.

Links

Topics: Dynamics Concepts: Newton’s second law · Tension Skills: Vector resolution Objects: Simple pendulum