A diving board length 5.60m that weighs 250N is supported by two pillars. One pillar is at the left end of the diving board, as shown below; the other is distance 1.55m away .
Find the magnitude of the force exerted by the left pillar when a 90.0kg diver stands at the far end of the board.
Find the magnitude of the force exerted by the right pillar when a 90.0kg diver stands at the far end of the board.
To find the magnitude of the force exerted by the left pillar, let's consider the torque equation.
Torque is the force applied to an object multiplied by the perpendicular distance from the axis of rotation to the line of action of the force. In this case, the axis of rotation is the right pillar, and the force is exerted by the left pillar.
The torque equation is given by τ = r * F * sin(θ), where:
- τ is the torque,
- r is the perpendicular distance from the axis of rotation to the line of action of the force,
- F is the magnitude of the force, and
- θ is the angle between the force vector and the line joining the axis of rotation and the point of application of the force.
In this case, the perpendicular distance from the axis of rotation (right pillar) to the line of action of the force (left pillar) is the length of the diving board, which is 5.60 m. The angle θ is 0 degrees because the force and the line joining the pillars are in the same direction.
So, the torque equation simplifies to τ = 5.60 m * F.
To find the magnitude of the force exerted by the left pillar, we need to balance the torques. The torque exerted by the diver can be calculated as τ_diver = m * g * d, where:
- m is the mass of the diver,
- g is the acceleration due to gravity (approximately 9.8 m/s^2), and
- d is the perpendicular distance from the far end of the diving board to the right pillar, which is 1.55 m.
Since the diving board is in equilibrium, the torques exerted by the diver and the left pillar must be equal, so τ_diver = 5.60 m * F.
Substituting the values, we get:
90.0 kg * 9.8 m/s^2 * 1.55 m = 5.60 m * F
Solving for F, we find:
F = (90.0 kg * 9.8 m/s^2 * 1.55 m) / 5.60 m
Now, we can calculate the magnitude of the force exerted by the left pillar.
To find the magnitude of the force exerted by the right pillar, we need to note that the sum of the vertical forces must be zero because the diving board is in equilibrium. So, the upward force exerted by the right pillar must balance the downward force exerted by the left pillar and the weight of the diver.
The magnitude of the force exerted by the right pillar can be calculated by summing the forces in the vertical direction. Let's assume the force exerted by the right pillar is denoted by F_right.
F_right - F_left - m * g = 0
Substituting the values, we get:
F_right - [(90.0 kg * 9.8 m/s^2 * 1.55 m) / 5.60 m] - (90.0 kg * 9.8 m/s^2) = 0
Solving for F_right, we can find the magnitude of the force exerted by the right pillar.
Therefore, the magnitude of the force exerted by the left pillar is [(90.0 kg * 9.8 m/s^2 * 1.55 m) / 5.60 m], and the magnitude of the force exerted by the right pillar is determined by solving the equation above.