The diving board shown in the figure below has a mass of 28 kg and its center of mass is at the board's geometrical center.
(Figure 1)
Determine the forces that support posts 1 and 2 (separated by 1.4 m) exert on the board when a 60-kg person stands on the end of the board 2.8 m from support post 2. Use positive values for the upward forces and negative values for the downward forces.
To determine the forces that support posts 1 and 2, we need to consider the gravitational forces acting on the diving board and the person standing on it.
Let's break down the problem step by step:
Step 1: Find the gravitational force acting on the diving board.
The gravitational force on an object can be calculated using the formula:
Force = mass x acceleration due to gravity
For the diving board, the gravitational force can be calculated as:
Force_board = mass_board x acceleration due to gravity
Given:
mass_board = 28 kg
acceleration due to gravity = 9.8 m/s^2 (approximately)
Substituting the values:
Force_board = 28 kg x 9.8 m/s^2 = 274.4 N
So, the gravitational force acting on the diving board is 274.4 N.
Step 2: Find the gravitational force acting on the person.
Similar to the diving board, the gravitational force on the person can be calculated using the same formula:
Force_person = mass_person x acceleration due to gravity
Given:
mass_person = 60 kg
Substituting the values:
Force_person = 60 kg x 9.8 m/s^2 = 588 N
So, the gravitational force acting on the person is 588 N.
Step 3: Calculate the total downward force acting on the diving board.
Since the person is standing on the end of the diving board, the weight of the person will exert a downward force on the diving board.
The total downward force on the diving board is the sum of the gravitational force of the board and the gravitational force of the person:
Total_downward_force = Force_board + Force_person
Substituting the values:
Total_downward_force = 274.4 N + 588 N = 862.4 N
So, the total downward force acting on the diving board is 862.4 N.
Step 4: Divide the total downward force between the two support posts.
Since the diving board is at equilibrium, the total downward force acting on it must be balanced by the vertical forces (upward forces) exerted by the support posts.
Let F1 represent the upward force exerted by support post 1 and F2 represent the upward force exerted by support post 2.
Since posts 1 and 2 are separated by 1.4 m, the force exerted by post 2 will be double the force exerted by post 1.
Therefore, we can write the following equation to represent the equilibrium condition:
F1 + 2(F1) = Total_downward_force
Simplifying the equation:
3F1 = 862.4 N
Dividing both sides by 3:
F1 = 862.4 N / 3 = 287.47 N
So, the upward force exerted by support post 1 is approximately 287.47 N.
Since support post 2 exerts twice the force of support post 1, the upward force exerted by support post 2 can be calculated as:
F2 = 2F1 = 2(287.47 N) = 574.94 N
So, the upward force exerted by support post 2 is approximately 574.94 N.
Therefore, the forces that support posts 1 and 2 exert on the board are approximately 287.47 N (upward) and -574.94 N (downward), respectively.