An 80 kg man stands in an elevator. What force does the floor of the elevator exert on the man if
a) the elevator is stationary?
b) the elevator accelerates upward at 2 m/s2?
c) the elevator rises with constant velocity 4 m/s?
d) while rising, the elevator decelerates at 1.5 m/s2?
e) the elevator descends with constant velocity 7 m/s?
Please show step by step with the equations for each one
Sure! Let's break down each scenario one by one, using Newton's second law: F = ma, where F is the force, m is the mass, and a is the acceleration.
a) When the elevator is stationary, there is no acceleration, so the force exerted by the floor on the man is equal to his weight. The weight can be calculated as the product of the mass (80 kg) and the acceleration due to gravity (9.8 m/s^2). So, the force in this case is 80 kg * 9.8 m/s^2 = 784 N.
b) When the elevator accelerates upward at 2 m/s^2, we need to consider both the weight of the person and the upward acceleration. To find the force, we need to calculate the net force acting on the man. The weight of the person is the same as before, 784 N. The acceleration due to gravity is always acting downward, so the net force in the upward direction is the sum of the gravitational force and the upward force due to acceleration. It can be calculated as follows:
Net force = ma (where m is the mass in kg, and a is the acceleration in m/s^2)
Net force = 80 kg * (9.8 m/s^2 + 2 m/s^2) = 80 kg * 11.8 m/s^2 = 944 N.
Therefore, the force exerted by the floor on the man with an upward acceleration of 2 m/s^2 is 944 N.
c) When the elevator rises with a constant velocity of 4 m/s, there is no acceleration. Hence, the force exerted by the floor on the man is simply his weight, which is 784 N.
d) When the elevator rises and decelerates at 1.5 m/s^2, we once again need to consider the weight of the person and the acceleration. This time, the acceleration is in the opposite direction as the weight. The net force will be the difference between the gravitational force and the downward force due to deceleration:
Net force = ma
Net force = 80 kg * (9.8 m/s^2 - 1.5 m/s^2) = 80 kg * 8.3 m/s^2 = 664 N.
Therefore, the force exerted by the floor on the man while the elevator decelerates at 1.5 m/s^2 is 664 N.
e) When the elevator descends with a constant velocity of 7 m/s, there is no acceleration. So, the force exerted by the floor on the man is again his weight, which is 784 N.
I hope this explanation helps! Let me know if you have any further questions.