a man is working on a plank which is placed horizontally across two support so that equal length project at each end.if the plank is 5m long and has a mass of 30kg what is the greatest distance from each end at which the support can be placed so that a man of mass 80kg can stand anywhere on the plank without him falling off the support
To solve this problem, we can use the concept of torque. Torque is the tendency of a force to rotate an object about an axis or pivot point.
Let's assume the distance from one end of the plank to the respective support is x meters. Since the plank is balanced, the torques on both sides must be equal.
The torque on each side of the plank is given by the formula: Torque = Force x Distance.
Considering the man's weight as the force acting on the plank, we can express the torque equation as follows:
Torque on the left side = Torque on the right side
(Mass of the man x Acceleration due to gravity x Distance from the left support) = (Mass of the plank x Acceleration due to gravity x Distance from the right support)
Using the given values:
(80 kg x 9.8 m/s² x x) = (30 kg x 9.8 m/s² x (5 - x))
Simplifying the equation:
784x = 294(5 - x)
784x = 1470 - 294x
1078x = 1470
x = 1.36 meters (rounded to two decimal places)
Therefore, the greatest distance from each end at which the supports can be placed, so that a man of mass 80 kg can stand anywhere on the plank without falling off, is approximately 1.36 meters.
To find the greatest distance from each end at which the support can be placed, we need to consider the conditions for stability. For the man not to fall off the support, the plank must remain in equilibrium.
One way to analyze this is by considering the torque exerted by the man's weight on the plank. Torque is equal to the force applied multiplied by the distance from the pivot point.
Let's assume that the distance between the supports is d.
For the man at the end of the plank, to avoid falling off, the torque exerted by his weight should not cause the plank to rotate.
The torque exerted by the man's weight can be calculated as follows:
Torque = (man's weight) * (distance from the pivot)
The torque exerted by the plank's weight can be calculated as follows:
Torque = (plank's weight) * (distance from the pivot)
Torque = (30 kg) * (0.5 * 5m) = 75 kgm²/s²
For equilibrium, the net torque should be zero.
Now let's calculate the torque exerted by the man's weight:
Torque = (80 kg) * (d)
Setting the torques equal to each other, we can solve for d:
80d = 75
d = 75/80
d ≈ 0.9375 meters
Therefore, the greatest distance from each end at which the support can be placed is approximately 0.9375 meters.