A 59 kg diver stands at the end of a 32 kg springboard, the spring board is 3 m long. The board is attached to a hinge at the left end, but simply rests on the right support which is half way along the board. What is the magnitude of the vertical force exerted by the hinge on the board?

The magnitude of the vertical force exerted by the hinge on the board is equal to the sum of the diver's weight and the weight of the springboard, or 91 kg.

To find the magnitude of the vertical force exerted by the hinge on the board, we need to analyze the forces acting on the system.

First, let's consider the weight of the diver. The weight is given by the formula: weight = mass * gravity, where mass is the mass of the diver and gravity is the acceleration due to gravity.

weight = 59 kg * 9.8 m/s^2 = 578.2 N

Next, we need to analyze the forces on the springboard. Since the springboard is simply resting on the right support, it exerts a reaction force equal to the weight of the diver and the springboard combined.

The total weight of the diver and the springboard is the sum of their individual weights:

total weight = weight of diver + weight of springboard

The weight of the springboard is given by: weight = mass * gravity

weight of springboard = 32 kg * 9.8 m/s^2 = 313.6 N

Therefore, the total weight is:

total weight = 578.2 N + 313.6 N = 891.8 N

Since the springboard is resting on the support and not attached to it, the vertical force exerted by the hinge on the board must be equal in magnitude and opposite in direction to the total weight.

Thus, the magnitude of the vertical force exerted by the hinge on the board is 891.8 Newtons.

To find the magnitude of the vertical force exerted by the hinge on the board, we need to analyze the forces acting on the system.

Let's consider the forces acting on the springboard:

1. Weight of the diver (acting downward):
- Since the diver's mass is 59 kg and acceleration due to gravity is approximately 9.8 m/s^2, the weight can be calculated as W = m * g = 59 kg * 9.8 m/s^2.

2. Weight of the springboard (acting downward):
- Since the springboard's mass is 32 kg and acceleration due to gravity is approximately 9.8 m/s^2, the weight can be calculated as W = m * g = 32 kg * 9.8 m/s^2.

3. Reaction force at the right support (acting upward):
- The right support exerts an upward force to balance the weight of the diver and the springboard.

4. Force applied by the hinge (unknown):
- The hinge applies a vertical force to balance the weight of the diver and the springboard.

Since the system is in equilibrium (not accelerating vertically), the sum of the vertical forces must be zero.

W(diver) + W(springboard) + Reaction force at right support + Force by hinge = 0

Substituting the weight formulas we derived earlier, we have:

(m(diver) * g) + (m(springboard) * g) + Reaction force at right support + Force by hinge = 0

Substituting the given values:

(59 kg * 9.8 m/s^2) + (32 kg * 9.8 m/s^2) + Reaction force at right support + Force by hinge = 0

Simplifying the equation:

(578.2 N) + (313.6 N) + Reaction force at right support + Force by hinge = 0

892.2 N + Reaction force at right support + Force by hinge = 0

Now, let's consider the torque (moment of force) acting on the system.

The torque created by the diver's weight wants to make the springboard rotate clockwise, while the torque created by the weight of the springboard wants to make the springboard rotate counterclockwise.

To balance the torques, the force applied by the hinge must create an equal and opposite torque.

The torque created by the weight of the diver can be calculated as T(diver) = (distance from hinge to diver) * (weight of diver)

The torque created by the weight of the springboard can be calculated as T(springboard) = (distance from hinge to center of springboard) * (weight of springboard)

Since the springboard is 3 m long and the right support is halfway along the board (1.5 m from the hinge), the distance from the hinge to the center of the springboard is 1.5 m.

Therefore:

T(diver) = (2 m) * (W(diver))
T(springboard) = (1.5 m) * (W(springboard))

To balance the torques, we have:

T(springboard) = T(diver)

Substituting the torque formulas we derived earlier, we have:

(1.5 m) * (W(springboard)) = (2 m) * (W(diver))

Substituting the weight formulas we derived earlier, we have:

(1.5 m) * (32 kg * 9.8 m/s^2) = (2 m) * (59 kg * 9.8 m/s^2)

Simplifying the equation:

(1.5 * 313.6 N) = (2 * 578.2 N)
470.4 N = 1156.4 N
470.4 N + 1156.4 N = 1626.8 N

The reaction force at the right support is 1626.8 N.

Since the sum of the vertical forces must be zero, the force exerted by the hinge can be calculated as:

Force by hinge = -(W(diver)) - (W(springboard)) - Reaction force at right support

Substituting the weight formulas we derived earlier and the given values:

Force by hinge = -(59 kg * 9.8 m/s^2) - (32 kg * 9.8 m/s^2) - (1626.8 N)

Force by hinge = -578.2 N - 313.6 N - 1626.8 N

Force by hinge = -2518.6 N

Since the force exerted by the hinge acts upward, the magnitude of the vertical force exerted by the hinge on the board is 2518.6 N.