A 50N diver stands at the end of a 4.0m diving board. The board is attached by two
supports 1.5 m apart as shown below. Find the tension in each of the two supports if the
diving board weighs 150N.
To find the tension in each of the two supports, we need to consider the forces acting on the diving board. Let's break down the problem step by step:
1. Determine the total weight of the diving board: The weight of the diving board is given as 150N. This weight acts downward, pulling the diving board towards the ground.
2. Determine the weight of the diver: The weight of the diver is given as 50N. This weight also acts downward, exerting a force on the diving board.
3. Identify the pivot point: In this case, the pivot point is where the diving board attaches to the supports.
4. Determine the distance between the weight of the diving board and the pivot point: The distance between the weight of the diving board and the pivot point is half the length of the diving board, which is 2.0m.
5. Determine the distance between the weight of the diver and the pivot point: The distance between the weight of the diver and the pivot point is the full length of the diving board, which is 4.0m.
6. Calculate the torque due to the weight of the diving board: Torque is given by the formula Torque = Force x Distance. In this case, the force is the weight of the diving board (150N) and the distance is 2.0m. So the torque due to the weight of the diving board is 150N x 2.0m = 300Nm.
7. Calculate the torque due to the weight of the diver: Torque is given by the formula Torque = Force x Distance. In this case, the force is the weight of the diver (50N) and the distance is 4.0m. So the torque due to the weight of the diver is 50N x 4.0m = 200Nm.
8. Since the supports are in equilibrium, the sum of the torques on one support should be equal to the sum of the torques on the other support.
9. Assume that the tension in the left support is T1 and the tension in the right support is T2. The torque due to T1 can be calculated by multiplying T1 by the distance between the pivot point and the left support, which is 1.5m. Similarly, the torque due to T2 can be calculated by multiplying T2 by the distance between the pivot point and the right support, which is also 1.5m.
10. Set up an equation: Since the torques on both supports should add up to the total torques due to the weight of the diving board and the weight of the diver, we can set up the equation: T1 x 1.5m + T2 x 1.5m = 300Nm + 200Nm.
11. Finally, solve the equation to find the tensions in the supports: Simplify the equation to 1.5T1 + 1.5T2 = 500Nm. Divide both sides of the equation by 1.5 to get T1 + T2 = 333.33Nm. Since the diving board is in equilibrium, the total tension in the supports is equal to the total weight (150N + 50N = 200N). So we have the equation T1 + T2 = 200N.
12. Now, we have two equations: T1 + T2 = 333.33Nm and T1 + T2 = 200N. Solve these equations simultaneously to find the values of T1 and T2.
13. Subtract the second equation from the first equation: 333.33Nm - 200N = 133.33Nm. This equation shows that the difference between the tensions in the two supports is 133.33Nm.
14. Since T2 is greater than T1, we can assign the difference to T2: T2 = 133.33Nm.
15. Substitute the value of T2 into the second equation: T1 + 133.33Nm = 200N. Subtract 133.33Nm from both sides: T1 = 200N - 133.33Nm = 66.67Nm.
So, the tension in the left support (T1) is 66.67Nm, and the tension in the right support (T2) is 133.33Nm.