A person is carrying a plank of wood 2.00 m long with one hand pushing down on it at one end
with a force F1 and the other hand holding it up at 50.0 cm from the same end of the plank with
force F2. If the plank has a mass of 20.0 kg and its center of gravity is at the middle of the plank,
what are the forces F1 and F2?
To find the forces F1 and F2, we can use the principle of torque equilibrium. Torque is the measure of a force's tendency to cause a rotation. When an object is in rotational equilibrium, the sum of the torques acting on it must be zero.
In this case, the plank is in rotational equilibrium, so the sum of the torques acting on it must be zero. The torque due to force F1 (pushing down) is equal to the torque due to force F2 (holding it up), which can be calculated using the equation:
Torque = Force * Distance
Let's calculate the torques and set them equal to each other.
Torque due to F1 = F1 * distance1
Torque due to F2 = F2 * distance2
Given:
Length of the plank (L) = 2.00 m
Mass of the plank (m) = 20.0 kg
Distance of F2 from the end (distance2) = 50.0 cm = 0.50 m
Since the center of gravity is at the middle of the plank, the distance1 is half the length of the plank.
distance1 = L/2 = 2.00 m / 2 = 1.00 m
Now we can set up the equation for torque equilibrium:
F1 * distance1 = F2 * distance2
Substituting the given values:
F1 * 1.00 m = F2 * 0.50 m
Now we can solve for F1.
F1 = (F2 * 0.50 m) / 1.00 m
Since the mass of the plank is given, we can also relate F1 and F2 to the weight of the plank.
Weight of the plank (W) = mass * acceleration due to gravity
W = m * g
So, the weight of the plank is:
W = 20.0 kg * 9.8 m/s² = 196 N
Since the plank is in equilibrium, the sum of the forces in the vertical direction must be zero. Therefore:
F1 + F2 = W
Now we have two equations:
F1 * 1.00 m = F2 * 0.50 m
F1 + F2 = 196 N
We can solve these equations simultaneously to find the values of F1 and F2.
Using substitution method, we substitute the value of F1 from the first equation into the second equation:
(F2 * 0.50 m) / 1.00 m + F2 = 196 N
Now simplify the equation:
0.50 F2 + F2 = 196 N
1.50 F2 = 196 N
F2 = 196 N / 1.50 = 130.67 N
Now substitute the value of F2 back into the first equation to find F1:
F1 * 1.00 m = 130.67 N * 0.50 m
F1 * 1.00 m = 65.34 N
F1 = 65.34 N
So, the force F1 is 65.34 N and the force F2 is 130.67 N.