Suppose a crate of mass 2.8 kg is placed on the plank in the figure below at a distance 5.1 m from the left end. Find the forces exerted by the two supports on the plank. (The mass of the plank is 12.0 kg.)
Left Support N
Right Support N
The plank set on two pivots one at 3.0m from the left, and the other 2.0m from the right. There is 7.0m between the two pivots
The sum of the two forces is
F1 + F2 = (2.8 + 12.0) g Newtons
To get F1 or F2, set the sum of the moments about either support equal to zero. The weight of the plank acts at its center of mass.
You should be able to write that equation by looking at the figure you have not provided.
To find the forces exerted by the two supports on the plank, we need to consider the torques acting on the plank due to the weight of the crate and the weight of the plank itself.
First, let's calculate the torque exerted by the crate. The weight of the crate can be calculated using the formula:
Weight = Mass x Gravity
where the mass of the crate is 2.8 kg and the acceleration due to gravity is approximately 9.8 m/s².
Weight of the crate = 2.8 kg x 9.8 m/s² = 27.44 N
The torque exerted by the crate can be calculated by multiplying the weight of the crate by its distance from the left pivot, 5.1 m.
Torque due to crate = Weight of the crate x Distance = 27.44 N x 5.1 m = 139.944 N·m
Next, let's calculate the torque exerted by the plank itself. The weight of the plank can be calculated using the same formula as before.
Weight of the plank = Mass of the plank x Gravity = 12.0 kg x 9.8 m/s² = 117.6 N
Since the pivot on the left is at a distance of 3.0 m from the left end of the plank, and the pivot on the right is 2.0 m from the right end, we can calculate the torque due to the plank by considering the torques about the left pivot and the right pivot separately.
Torque due to plank about left pivot = Weight of the plank x Distance = 117.6 N x 3.0 m = 352.8 N·m
Torque due to plank about right pivot = Weight of the plank x Distance = 117.6 N x 2.0 m = 235.2 N·m
Now let's determine the net torque acting on the plank. Since the plank is in equilibrium, the net torque must be zero.
Net torque = Torque due to crate + Torque due to plank about left pivot - Torque due to plank about right pivot = 139.944 N·m + 352.8 N·m - 235.2 N·m
Simplifying, we get:
Net torque = 257.544 N·m
Finally, let's find the forces exerted by the two supports on the plank. The left support exerts a force perpendicular to the plank, and the right support exerts a force parallel to the plank. These forces can be calculated using the equation:
Net torque = Force x Lever Arm
where the lever arm is the distance between the pivot and the line of action of the force.
For the left support:
Force of the left support x Distance = Net torque
Force of the left support x 3.0 m = 257.544 N·m
Force of the left support = 257.544 N·m / 3.0 m = 85.848 N
For the right support:
Force of the right support x Distance = Net torque
Force of the right support x 2.0 m = 257.544 N·m
Force of the right support = 257.544 N·m / 2.0 m = 128.772 N
Therefore, the forces exerted by the two supports on the plank are approximately 85.848 N for the left support and 128.772 N for the right support.