A mass of 108 g is hanging from two massless ropes attached to the ceiling.
One rope makes an angle of 50° with the ceiling, while the other makes an angle of 29°.
Find the tensions in the two ropes.
Let's begin by drawing our mass hanging from the two ropes:
Looking at our sketch, we can infer that the mass is subject to 3 forces:
Here's the free-body diagram of our hanging mass:
We know the mass (108 g, which in kilograms is 0.108 kg), and the angles that the two ropes make with the ceiling (50° and 29°).
We are asked to find the tensions in the two ropes.
The mass is hanging.
What does this tell us?
This tells us that the acceleration of the mass must be zero.
And because the acceleration is zero, the resultant force acting on the mass is also zero. Indeed, for Newton's 2nd Law:
Now that we determined that the resultant force acting on the mass is zero, we can find the tensions of the two ropes using the following step by step process:
Let's start with the first step.
We draw the coordinate axes on our free-body diagram. For convenience, we choose the x axis horizontal and the y axis vertical. Then, we determine the x and y components of all the forces that act on the mass.
We need to keep in mind that the angle between each tension force and its x component is equal to the angle that the rope, producing that tension, makes with the ceiling:
Thus, the x and y components of the resultant force will be:
Using these two equations we can easily find T1 and T2.
There are multiple ways in which we can do this. One way would be to first solve Eq. (3) for T2:
|T2 = T1||cos 50°|
Then, we substitute T2 with 0.735T1 in Eq. (4):
And we solve it for T1:
|T1 =||(0.108 kg) (9.81 N/kg)|
Finally, we substitute the value of T1 in Eq. (5) to find T2:
Therefore, the tension in the rope at the 50° angle is 0.946 N, and the tension in the rope at the 29° angle is 0.695 N.