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### 402-H: Elevators 1: Drawing a Free-Body Diagram Given Acceleration of an Elevator

• Topic Cluster: Dynamics
• Topic: Quantitative Dynamics (Newton's Second Law)
• Objective: Given a person in an elevator with a particular mass and acceleration, draw a correct free-body diagram of that person.
• Content: Inside an elevator, gravity and normal force act on the rider; normal force is a constraint force that adjusts to match a particular acceleration
• Level: 3

#### BACK to Ladder Quantitative Dynamics (Newton's Second Law)

In levels 1 and 2, we have learned to use free-body diagrams and Newton's Second Law. Here on level 3, we will use problems where you need to identify individual forces using the principles of those forces. For these problems, regarding a person inside of an elevator, there are only two forces involved: gravity and normal force, but you need to understand each force in order to use it.

You may have completed some of the block dragging problems already. In these problems, the magnitude normal force is equal to the magnitude of gravitational force of the block. However, this is not always the case. Rather, there is a procedure for finding the magnitude of normal force that requires you to know the net force first. In these problems, we will begin using that procedure.

#### Forces Involved

For someone riding an elevator, there are always two forces acting on them: The force of gravity is acting downward and the normal force is acting upward. Gravity, always has the same force, but the normal force acn be greater or lesser, so it looks something like this:

• #### The Force of Gravity

• On earth, the direction of gravity is always down.
• On earth, the magnitude of gravity is given by the formula $$F_g = m g$$.

$$F_g = mg$$

Symbol
Quantity
SI Unit
Notes
$$F_{g}$$
Magnitude of kinetic friction
Newtons (N)
$$m$$
mass
kilograms (kg)
$$g$$
gravitational field / free-fall acceleration
m/s^2
on the surface of earth, is equal to 9.81 m/s^2
• #### Normal Force

In these problems, the direction of normal force is always up. To find the magnitude of normal force, there is no set formula. Instead, you need to follow a two-step process:
1. First, find the net force.
2. Then, find the normal force using the process of finding the missing force.

#### Other Principles

1. Newton's Second Law is crucial towards understanding these elevator problems. Note that the direction of the net force is always the same as the direction of the acceleration.

$$\Sigma F = m a$$

2. Newton's First Law states that whenever an object is not moving or moving at a constant velocity, the net force acting on that object is zero.
1. Esther is standing in an elevator. Ester has a mass of 80.0 kg. The elevator is moving downwards and is accelerating upwards at a rate of 0.200 m/s2.
1. What is the magnitude and direction of net force acting on Esther?
2. Edgar is standing in an elevator. Edgar has a mass of 70.0 kg. The elevator is moving downwards and is accelerating downwards at a rate of 0.400 m/s2.
1. What is the magnitude and direction of net force acting on Edgar?
3. Esther is standing in an elevator. Esther has a mass of 64.0 kg. The elevator is moving downwards and is accelerating downwards at a rate of 0.900 m/s2.
1. What is the magnitude and direction of net force acting on Esther?
4. Ezekiel is in the elevator. Ezekiel has a mass of 68.0 kg. The elevator is moving upwards and accelerating upwards at a rate of 0.800 m/s2.
1. What is the magnitude and direction of net force acting on Ezekiel?
5. Eddie is standing in an elevator, and the elevator is not moving. Eddie has a mass of 70.0 kg.
1. What is the net force acting on Eddie? How do you know?
6. Evelyn is standing in an elevator. It is moving upward at a constant velocity. Evelyn has a mass of 60.0 kg.
1. What is the net force acting on Evelyn? How do you know?