A man lies on a plank of negligible mass supported by two scales 2.50 m apart. His head rests on the left scale which is also the axis of rotation. If the left scale reads 290 N and the right scale reads 122 N calculate his mass.
To calculate the man's mass, we need to use the principles of torque and rotational equilibrium.
First, let's analyze the forces acting on the plank and the man lying on it. There are two forces acting on the plank: the weight of the man and the normal force from the scales. Since the plank is not rotating, the net torque must be zero.
The torque (τ) is given by the formula τ = F x r, where F is the force applied and r is the distance from the axis of rotation at which the force is applied.
In this case, the left scale provides the axis of rotation. Therefore, the torque from the right scale is zero since the distance from the axis of rotation is zero. The only torque acting on the plank is due to the weight of the man, which is acting downward at a distance of 2.50 m away from the left scale.
The equation for torque can be written as:
τ = (force on left scale) x (distance from axis of rotation to force on left scale) - (force on right scale) x (distance from axis of rotation to force on right scale)
By substituting the given values into the equation:
τ = (290 N) x (2.50 m) - (122 N) x (0 m)
τ = 725 N·m
Since the net torque must be zero, the torque from the man's weight must be equal in magnitude but opposite in direction. Therefore:
τ = - (force of gravity on the man) x (distance from axis of rotation to the man's head)
By rearranging the equation and solving for the force of gravity on the man:
(force of gravity on the man) = - (τ / distance from axis of rotation to the man's head)
(force of gravity on the man) = - (725 N·m / 2.50 m)
(force of gravity on the man) = - 290 N
Since the force of gravity on the man is equal in magnitude to his weight, the man's weight is 290 N. Weight (W) is given by the formula W = m x g, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s^2).
By substituting the values into the equation:
290 N = m x (9.81 m/s^2)
Solving for m, we get:
m = 290 N / (9.81 m/s^2)
m ≈ 29.6 kg
Therefore, the man's mass is approximately 29.6 kg.