You know that you can safely stand on the hang-over end of a heavy plank that rests on a table. How far depends on your mass and the mass of the plank. Suppose you can stand on the end of a plank that overhangs the edge of the supporting table 1/5 its total length. Then how massive is the plank compared to your mass?
a) 1 and 1/3 times
b) more than 1 and 1/3 times
c) 2/3
d) the same
e) 1/3
To determine the mass of the plank compared to your mass, we need to use the principle of torque or rotational equilibrium.
Torque is defined as the product of the force applied and the distance from the point of rotation. In this case, the point of rotation is the edge of the table, and the force applied is your weight.
Let's assume your mass is m, and the mass of the plank is M. The distance from the edge of the table to the center of mass of the plank is L/2, where L is the total length of the plank. According to the problem, the overhang distance is 1/5 of the total length of the plank, so the overhang length is L/5.
Now, to find the torque exerted by the plank, we calculate the force exerted by the plank on the table. This force is equal to the weight of the plank, which is M multiplied by the acceleration due to gravity (g).
Since the plank is in rotational equilibrium, the torque exerted by the plank must be equal and opposite to the torque exerted by your weight.
The torque exerted by your weight is equal to your weight (m x g) multiplied by the overhang length (L/5).
The torque exerted by the plank is equal to the force exerted (M x g) multiplied by the distance from the edge of the table to the center of mass of the plank (L/2).
From this, we can set up an equation:
(m x g) x (L/5) = (M x g) x (L/2)
Dividing both sides by (M x g):
(m x (L/5)) / (L/2) = M
Simplifying this equation:
m/5 = M
Therefore, the mass of the plank is five times your mass. So, the correct option is:
a) 1 and 1/3 times