a person stands a distance of 0.300 meters from the right end of a 2.00 meter long uniform platform that is supported by two posts, one at each end. The board has a mass of 20.0 kg and the person's mass is 75.0 kg.
a. determine the force exterted on the board by the post on the left end.
b. determine the force exerted on the board by the post on the right end.
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Sure! I'd be happy to help you with these problems.
To determine the forces exerted by the posts on the board, we can start by considering the equilibrium of the board. In equilibrium, the sum of the forces acting on the board in the horizontal direction must be equal to zero.
Let's start with the force exerted by the post on the left end (denoted as F_left). Since this force acts towards the right, we will consider it positive. We can use the equation:
F_left - F_right = 0,
where F_right is the force exerted by the post on the right end (which we'll solve for in part b).
For part a, we need to determine F_left. To do this, let's consider the torques acting on the board.
The torque exerted by the person's weight (mg) about the left end of the board is given by:
Torque_person = mg * d_person,
where m is the mass of the person (75.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and d_person is the distance from the left end to the person (0.300 m).
Similarly, the torque exerted by the board's weight (m_board * g) about the left end of the board is given by:
Torque_board = m_board * g * d_board/2,
where m_board is the mass of the board (20.0 kg) and d_board is the length of the board (2.00 m).
For the board to be in equilibrium, the torques exerted by the person and the board must cancel each other out. Therefore, we have:
Torque_person = Torque_board,
mg * d_person = m_board * g * d_board/2.
Plugging in the given values, we can solve for d_board:
(75.0 kg * 9.8 m/s^2) * (0.300 m) = (20.0 kg * 9.8 m/s^2) * (d_board/2).
Now, solving for d_board gives us:
(75.0 kg * 9.8 m/s^2 * 0.300 m) = (20.0 kg * 9.8 m/s^2) * (d_board/2),
d_board = (75.0 kg * 0.300 m) / (20.0 kg) = 1.125 m.
Now that we know the distance from the left end to the center of the board (d_board/2 = 1.125 m/2 = 0.5625 m), we can determine F_left by setting the torques equal to each other and solving for F_left:
(m_board * g * d_board/2) = (m_person * g * d_person) + (F_left * d_board/2),
(20.0 kg * 9.8 m/s^2 * 1.125 m/2) = (75.0 kg * 9.8 m/s^2 * 0.300 m) + (F_left * 1.125 m/2).
Now, we can solve for F_left:
(20.0 kg * 9.8 m/s^2 * 1.125 m/2) - (75.0 kg * 9.8 m/s^2 * 0.300 m) = F_left * 1.125 m/2,
F_left = ((20.0 kg * 9.8 m/s^2 * 1.125 m/2) - (75.0 kg * 9.8 m/s^2 * 0.300 m)) / (1.125 m/2).
Calculate the expression on the right-hand side of the equation to find F_left.