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 = 12.1°. number i i units 6.79e13 save for later 1 using multiple attempts v 20% score reduction after -691.47 7.92e15 4845.4 250 attempts: 0 of 3 used submit answer

Explanation:

Step1: Analyze forces on sphere

Let the mass of the sphere be $m$. In the vertical - direction, the net force is zero since there is no acceleration in the vertical direction. So, $T\cos\theta=mg$, where $T$ is the tension in the string and $g = 9.8\ m/s^{2}$ is the acceleration due to gravity. In the horizontal direction, the net force provides the acceleration of the sphere (which is the same as the acceleration of the van), so $T\sin\theta=ma$.

Step2: Eliminate tension $T$

Divide the horizontal - force equation by the vertical - force equation: $\frac{T\sin\theta}{T\cos\theta}=\frac{ma}{mg}$. This simplifies to $\tan\theta=\frac{a}{g}$.

Step3: Solve for acceleration $a$

We know that $\theta = 12.1^{\circ}$ and $g = 9.8\ m/s^{2}$. Rearranging the equation $\tan\theta=\frac{a}{g}$ gives $a = g\tan\theta$. Substituting the values, we have $a=9.8\times\tan(12.1^{\circ})$.
$a = 9.8\times0.214 = 2.1\ m/s^{2}$

Answer:

$2.1\ m/s^{2}$