A uniform wooden plank of length L = 6.0 m and mass M = 90 kg rests on top of two sawhorses separated by D = 1.5 m, located equal distances from the center of the plank. You try to stand on the right-hand end of the plank. If the plank is to remain at rest, how massive can you be?
m = M x D / L - D
m = 90 x 1.5 / 6 - 1.5
m = 30 Kg
To determine the maximum mass that you can have without causing the plank to tip over, we need to consider the torques acting on the plank.
The torque caused by the gravitational force acting on the plank is given by the equation:
Torque_gravity = weight * perpendicular distance
The weight of the plank can be calculated using the formula:
Weight = mass * acceleration due to gravity
Since the length of the plank is 6.0 m and it is supported by sawhorses separated by 1.5 m, the perpendicular distance for the torque calculation is 1.5 m.
So, the torque caused by the weight of the plank is:
Torque_gravity = (mass * acceleration due to gravity) * perpendicular distance
Torque_gravity = (90 kg * 9.8 m/s^2) * 1.5 m
The torque caused by your weight (standing at the right-hand end of the plank) must be counteracted by the torque caused by the weight of the plank.
To determine the maximum mass you can have, we need to ensure that the sum of the torques is equal to zero in order for the plank to remain at rest.
Assuming you are at the very end of the right-hand side of the plank, the perpendicular distance from you to the center of the plank is 3.0 m.
Hence, the torque caused by your weight can be calculated as:
Torque_you = (your mass * acceleration due to gravity) * perpendicular distance
Torque_you = (your mass * 9.8 m/s^2) * 3.0 m
Since the plank is at rest, the sum of torques must be equal to zero:
Torque_gravity + Torque_you = 0
Substituting in the known values:
(90 kg * 9.8 m/s^2) * 1.5 m + (your mass * 9.8 m/s^2) * 3.0 m = 0
Simplifying:
1323 kg*m^2/s^2 + (your mass * 9.8 m/s^2 * 3.0 m) = 0
Solving for your mass:
9.8 m/s^2 * 3.0 m * your mass = -1323 kg*m^2/s^2
your mass = -1323 kg*m^2/s^2 / (9.8 m/s^2 * 3.0 m)
your mass = -45 kg*m / (9.8 m/s^2)
Therefore, the maximum mass you can have without causing the plank to tip over is approximately 4.6 kg.
To determine how massive you can be while still keeping the plank at rest, you need to calculate the torque on the plank.
The torque caused by your weight will try to rotate the plank in a clockwise direction, and it needs to be balanced by an equal and opposite torque to keep the plank at rest.
To calculate the torque, you can use the equation:
Torque = force x lever arm
For the plank to remain at rest, the torque caused by your weight should be balanced by the torque caused by the plank's weight.
The torque caused by your weight is given by:
Torque_you = (your weight) x (lever arm)
The torque caused by the plank's weight is given by:
Torque_plank = (plank's weight) x (lever arm)
= (mass of plank) x (gravitational acceleration) x (lever arm)
Since the plank is in equilibrium, the torques caused by your weight and the plank's weight must be equal:
Torque_you = Torque_plank
Now let's substitute the given values and solve for your weight (mass):
(your weight) x (lever arm) = (mass of plank) x (gravitational acceleration) x (lever arm)
mass_of_plank = (your weight) / (gravitational acceleration)
To find the maximum mass of a person who can stand on the right-hand end without tipping the plank, we need to find the maximum value for the mass of the person.
Let's assume that the right-hand end of the plank is at a distance (x) from the right sawhorse. Then the lever arm for your weight is (D - x).
Mass_of_person x (D - x) = mass_of_plank x D
Simplifying the equation, we have:
mass_of_person = (mass_of_plank x D) / (D - x)
Substituting the given values:
mass_of_person = (90 kg x 1.5 m) / (1.5 m - x)
So, the maximum mass you can have to keep the plank at rest depends on the position (x) of the right-hand end of the plank.
90 kg at 1.5 / 2 meters from center of plank exactly balanced your mass m at 3 meters from center
90 * 1.5 / 2 = m * 3
45 * 1.5 = 3 m
m = 22.5 kg