A uniform beam of weight 245 N, and length L, sticks out from a vertical wall. A lightweight cable connects the end of the beam to the wall, making an angle of 65.0° between the beam and the cable. What is the tension in the cable?
135 N
To find the tension in the cable, we can analyze the forces acting on the beam.
Let's denote the tension in the cable as T.
1. Resolve the weight of the beam into vertical and horizontal components:
- Vertical component: Wv = Weight * cos(65°)
- Horizontal component: Wh = Weight * sin(65°)
2. Now, consider the equilibrium of forces acting on the beam:
- In the vertical direction, the net force is zero since the beam is not moving vertically. Therefore, the vertical component of the tension must balance the vertical component of the weight:
T * cos(65°) = Wv
3. In the horizontal direction, there is no horizontal acceleration, so the net force is zero. Thus, the horizontal component of the tension must balance the horizontal component of the weight:
T * sin(65°) = Wh
4. Substitute the values we found for Wv and Wh:
T * cos(65°) = Weight * cos(65°)
T * sin(65°) = Weight * sin(65°)
5. Solve for T by dividing both equations by cos(65°) and sin(65°), respectively:
T = Weight * cos(65°) / cos(65°)
T = Weight * sin(65°) / sin(65°)
6. Simplify the equation:
T = Weight
So, the tension in the cable is equal to the weight of the beam, which is 245 N.
To find the tension in the cable, we can analyze the forces acting on the beam and use the principles of equilibrium.
Let's break down the forces acting on the beam:
1. Weight of the beam: This force acts vertically downward and has a magnitude of 245 N.
2. Tension in the cable: This force acts along the cable, pulling the beam towards the wall.
3. Normal force: This force is exerted by the wall on the beam, perpendicular to the wall. Since the beam is in equilibrium, the normal force must balance the component of the weight of the beam perpendicular to the wall.
In order to find the tension in the cable, we need to resolve the forces into their vertical and horizontal components, using the given angle of 65.0° between the beam and the cable.
Step 1: Resolve the weight of the beam into its vertical and horizontal components.
- The vertical component of the weight is given by Wv = Weight * cos(angle).
Wv = 245 N * cos(65.0°).
Step 2: Calculate the normal force exerted by the wall.
- The normal force balances the vertical component of the weight, so it is equal in magnitude but opposite in direction, N = -Wv.
Step 3: Resolve the weight of the beam into its horizontal component.
- The horizontal component of the weight is given by Wh = Weight * sin(angle).
Wh = 245 N * sin(65.0°).
Step 4: Set up an equation for equilibrium in the horizontal direction.
- The tension in the cable, T, balances the horizontal component of the weight, so T = Wh.
Therefore, the tension in the cable, T, is equal to the horizontal component of the weight of the beam, which is T = 245 N * sin(65.0°).
Calculating T:
T = 245 N * sin(65.0°)
T = 227.62 N (rounded to two decimal places)
Hence, the tension in the cable is approximately 227.62 N.