An escalator is moving downward with a uniform speed of u. A man of mass m is running upward on it at a uniform speed of v. If the height of the escalator is h, the work done by the man in going up the escalator is?
To find the work done by the man in going up the escalator, we need to first calculate the net velocity of the man with respect to the ground.
The velocity of the escalator is downwards with speed u, and the man is running upwards with speed v. So, the relative velocity of the man with respect to the escalator is (v + u).
Since the man is moving in the same direction (upwards) as the escalator is moving (downwards), the net velocity of the man with respect to the ground is given by:
v_net = v + u
Now, since the work done is equal to the force applied multiplied by the distance traveled, we need to calculate the force applied by the man.
The force applied is equal to the product of the mass of the man (m) and the acceleration (a). The net force acting on the man is the force of gravity (mg) minus the force provided by the escalator (m * (-u)).
So, the net force applied by the man is:
F_net = mg - m * (-u)
= mg + mu
Next, we need to calculate the distance traveled by the man.
The distance traveled is equal to the height of the escalator (h).
Now, we can calculate the work done by the man.
Work done = force applied * distance traveled
= F_net * h
= (mg + mu) * h
Therefore, the work done by the man in going up the escalator is (mg + mu) * h.
To find the work done by the man in going up the escalator, we need to understand the concept of work and the formula used to calculate it.
Work is defined as the product of the force applied to an object and the displacement of the object in the direction of the force. In this case, the force is the force exerted by the man to overcome gravity and move upward, and the displacement is the vertical distance covered by the man.
The work done by the man can be calculated using the formula:
Work = Force x Displacement
The force exerted by the man can be calculated using the equation:
Force = mass x acceleration
Acceleration, in this case, is the net acceleration of the man, which is the difference between his upward acceleration and the acceleration due to gravity acting downward.
Let's break down the situation and calculate the different components.
Given:
Mass of the man = m
Speed of the escalator = u
Speed of the man = v
Height of the escalator = h
To find the net acceleration of the man, we need to determine the individual accelerations acting on him.
1. Acceleration due to gravity:
The acceleration due to gravity, denoted by g, is a constant value which is approximately 9.8 m/s^2. Since the man is moving upward, the acceleration due to gravity acts in the opposite direction, downward. So, the acceleration due to gravity is -g.
2. Acceleration due to the escalator:
Since the escalator is moving downward with a uniform speed of u, the man, while moving upward, is also experiencing a downward acceleration equal to the acceleration of the escalator. So, the acceleration due to the escalator is -u.
Now, let's calculate the net acceleration of the man.
Net acceleration = upward acceleration - downward acceleration
= 0 - (-g - u)
= g + u
Now that we have the net acceleration, we can calculate the force exerted by the man.
Force = mass x acceleration
= m x (g + u)
The displacement of the man is the height of the escalator, h.
Finally, we can calculate the work done by the man.
Work = Force x Displacement
= [m x (g + u)] x h
Therefore, the work done by the man in going up the escalator is m x (g + u) x h.