A wrecking ball (weight 4140 N) is supported by a boom, which may be assumed to be uniform and has a weight of 2550 N. As the drawing shows, a support cable runs from the top of the boom to the tractor. The angle between the support cable and the horizontal is 32 degrees, and the angle between the boom and the horizontal is 48 degrees. Find the tension in the support cable and find the magnitude of the force exerted on the lower end of the boom by the hinge.

How do I take into account the boom for finding tension? Should I just sum the forces of the support cable in the x and y direction and subtract the force of the wrecking ball?

Sum forces about the top of the boom, sum horizontal components=0, sum vertical forces = 0. Then sum moments about the base of the boom. That should do it.

To find the tension in the support cable, you can follow these steps:

1. Resolve the force of the support cable into its horizontal and vertical components based on the given angle between the support cable and the horizontal (32 degrees). Let's call the tension in the support cable Tc.
- The horizontal component of the tension, Th, can be calculated as Th = Tc * cos(32°).
- The vertical component of the tension, Tv, can be calculated as Tv = Tc * sin(32°).

2. Sum the horizontal forces acting on the boom:
- The horizontal component of the tension (Th) is acting to the left.
- The weight of the boom (2550 N) is acting to the right.
- Therefore, the sum of the horizontal forces is Th - 2550 N = 0 (since the net horizontal force is zero).

3. Sum the vertical forces acting on the boom:
- The vertical component of the tension (Tv) is acting upward.
- The weight of the boom (2550 N) is acting downward.
- The weight of the wrecking ball (4140 N) is also acting downward.
- Therefore, the sum of the vertical forces is Tv - 2550 N - 4140 N = 0 (since the net vertical force is zero).

4. Solve the above two equations simultaneously to find the values of Th and Tv. Once you have the values of Th and Tv, you can use them to find the tension in the support cable (Tc) using the equation Tc = √(Th^2 + Tv^2).

To find the magnitude of the force exerted on the lower end of the boom by the hinge, you need to consider the moment about the base of the boom caused by the weight of the boom and the wrecking ball.

1. Calculate the moment caused by the weight of the boom (2550 N) about the base of the boom:
- Moment = Weight * Distance
- Distance can be determined using trigonometry as Distance = Length of the boom * sin(48°).
- Calculate the moment caused by the weight of the boom.

2. Calculate the moment caused by the weight of the wrecking ball (4140 N) about the base of the boom:
- The weight of the wrecking ball acts at the end of the boom, so its moment is zero.

3. The magnitude of the force exerted on the lower end of the boom by the hinge can be calculated as the sum of the moments about the base of the boom divided by the length of the boom.

By following these steps, you can determine the tension in the support cable and the magnitude of the force exerted on the lower end of the boom by the hinge.

noob