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 1/4 its total length. Then how massive is the plank compared to your mass?
a. 1/2 your mass
b. same as your mass
c. 1 1/2 your mass
d. twice your mass
e. 4 times your mass
any help would be appreciated!
To determine the mass of the plank compared to your mass, we need to consider the principle of equilibrium. When the plank is balanced, the torques on both sides of the pivot (the edge of the table) will be equal.
Let's assume your mass is M and the mass of the plank is P.
The torque exerted by your mass at the end of the plank can be calculated as follows:
Torque exerted by your mass = M * (1/4) * (1/2)
The torque exerted by the plank can be calculated as the product of its own mass and half of its length:
Torque exerted by the plank = P * (1/2) * (3/4)
Since the torques on both sides of the pivot are equal, we can equate these two expressions:
M * (1/4) * (1/2) = P * (1/2) * (3/4)
Simplifying the equation, we get:
M/8 = (3/2)P/8
M = 3P/2
Therefore, the mass of the plank is 1.5 times your mass.
So, the answer is option c. 1 1/2 your mass.
To solve this problem, we can use the concept of torque, which is the ability of a force to rotate an object around an axis. In this case, the torque exerted by your weight, which depends on your mass, should be balanced by the torque exerted by the plank and the overhanging part of the plank.
Let's denote the total length of the plank as L, and the length of the overhanging part as x. The distance between the pivot point (the edge of the table) and the center of mass of the plank is L/2, while the distance between the pivot point and your position is (L/4 + x).
The torque exerted by your weight is given by the formula:
Torque = mass * distance * gravity
The torque exerted by the plank and the overhang can be represented using the center of mass of the plank, which is at L/2. The torque exerted by this mass is given by:
Torque = (mass of the plank) * (distance) * gravity
Since the torques exerted by your weight and the plank must balance each other, we can set up an equation:
(mass of the plank) * (L/4 + x) = (your mass) * (L/2)
Now let's solve for the mass of the plank:
(mass of the plank) = [(your mass) * (L/2)] / (L/4 + x)
Simplifying the equation:
(mass of the plank) = 2 * (your mass) / (1 + 4x/L)
We are given that x = L/4, so:
(mass of the plank) = 2 * (your mass) / (1 + 4(L/4) / L)
= 2 * (your mass) / (1 + 1)
= 2 * (your mass) / 2
= your mass
Therefore, the mass of the plank is the same as your mass. The answer is (b) same as your mass.