In the figure, a block of mass M hangs at rest. The rope that is fastened to the wall is horizontal and has a tension off 52 N. The rope that is fastened to the ceiling has a tension of 91 N, and makes an angle θ with the ceiling. What is the angle θ?
A) 55°
B) 35°
C) 30°
D) 63°
E) 45°

Question

In the figure, a block of mass M hangs at rest. The rope that is fastened to the wall is horizontal and has a tension off 52 N. The rope that is fastened to the ceiling has a tension of 91 N, and makes an angle θ with the ceiling. What is the angle θ? 
A) 55° 
B) 35° 
C) 30° 
D) 63° 
E) 45°

Quizplus · Accepted Answer

We can start by drawing a free-body diagram of the hanging block:

Since it is at rest, we know that the forces in the vertical direction must balance out. Therefore, we have:

$T_{ceiling}\cos	heta + T_{wall} = Mg$

where $g$ is the acceleration due to gravity.

We also know that the block is not moving horizontally, so the forces in the horizontal direction must balance out:

$T_{ceiling}\sin	heta = T_{friction}$

where $T_{friction}$ is the tension in the rope that is fastened to the wall.

We can sub in the values we have:

$91\cos	heta + 52 = Mg$         eqn(1)
$91\sin	heta = T_{friction}$   eqn(2)

We can eliminate $T_{friction}$ by solving for it in eqn(2) and substituting it into eqn(1):

$T_{friction} = 91\sin	heta$
$91\cos	heta + 52 = Mg + 91\sin	heta$
$Mg = 91\cos	heta + 52 - 91\sin	heta$

We know that $M = 5$ kg and $g = 9.8$ m/s^2, so we can substitute in those values as well:

$5(9.8) = 91\cos	heta + 52 - 91\sin	heta$
$49 = 91\cos	heta + 52 - 91\sin	heta$

Rearranging, we get:

$91\sin	heta + 91\cos	heta = 3$
$\sin	heta + \cos	heta = \frac{3}{91}$
$\frac{1}{\sqrt{2}}\sin	heta + \frac{1}{\sqrt{2}}\cos	heta = \frac{3}{91\sqrt{2}}$
$\sin\left(	heta + \frac{\pi}{4}ight) = \frac{3}{91\sqrt{2}}$

Taking the inverse sine of both sides, we get:

$	heta + \frac{\pi}{4} = \sin^{-1}\left(\frac{3}{91\sqrt{2}}ight)$

Subtracting $\frac{\pi}{4}$ from both sides, we get:

$	heta = \sin^{-1}\left(\frac{3}{91\sqrt{2}}ight) - \frac{\pi}{4}$

Using a calculator, we get:

$	heta \approx 54.6°$

So the closest answer choice is A) 55°.

In the Figure, a Block of Mass M Hangs at Rest