Answer to Question #311855 in Mechanics | Relativity for Chen

Question

Two identical 15.0-kg balls, each 25.0 cm in diameter, are suspended by two

35.0-cm wires. The entire apparatus is supported by a single 18.0-cm wire,

and the surfaces of the balls are perfectly smooth.

a) Find the tension in each of the three wires.

b) How hard does each ball push on the other one?

Answer 1

Explanation & Calculations

a)

• Take the system as a whole so that only the two weights of the ball and the single thread is seen, and consider the equilibrium.

$\qquad\qquad \begin{aligned} \small T-2mg&=\small 0\\ \small T&=\small 294\,N \end{aligned}$

• If the two threads make $\small \theta$ angles with the horizontal (they are identical as the system is symmetrical about the vertical)

$\qquad\qquad \begin{aligned} \small \cos\theta &=\small \frac{r}{r+35.0}=\frac{12.5}{12.5+35.0}=0.263\\ \small \sin\theta &=\small 0.945 \end{aligned}$

• Now consider the equilibrium of a ball under the act of its weight and one of the two threads.

$\qquad\qquad \begin{aligned} \small \uparrow\\ \small T_1\sin\theta-mg&=\small 0\\ \small T_1&=\small 155.6\,N \end{aligned}$

b)

• Consider the horizontal equilibrium of a ball under the same act and say the force the balls are being pushed is R,

$\qquad\qquad \begin{aligned} \small \to\\ \small T_1\cos \theta-R&=\small 0\\ \small R&=\small 40.9\,N \end{aligned}$