A 69-kg man standing on a scale in an elevator notes that as the elevator rises, the scale reads 823 N. What is the acceleration of the elevator?
m/s2 upward
To determine the acceleration of the elevator, we need to consider the forces acting on the man. In this case, we have the force of gravitational pull acting downward (weight), and the force from the scale acting upward.
First, let's calculate the weight of the man. The weight can be determined by multiplying the mass (69 kg) by the acceleration due to gravity (9.8 m/s^2):
Weight = mass x acceleration due to gravity
Weight = 69 kg x 9.8 m/s^2
Weight = 676.2 N
Since the man is standing on the scale, the normal force exerted by the scale is equal in magnitude but opposite in direction to the weight. So we have:
Normal force = -Weight = -676.2 N
The scale reading (823 N) is the sum of the normal force and any additional force on the man due to the elevator's acceleration. Therefore, we have:
823 N = Normal force + additional force
To find the additional force, we subtract the normal force from the scale reading:
Additional force = scale reading - normal force
Additional force = 823 N - (-676.2 N)
Additional force = 823 N + 676.2 N
Additional force = 1499.2 N
Now, we can find the acceleration of the elevator by dividing the additional force by the mass of the man:
Acceleration = additional force / mass
Acceleration = 1499.2 N / 69 kg
Acceleration ≈ 21.7 m/s^2
So the acceleration of the elevator is approximately 21.7 m/s^2 upward.
To find the acceleration of the elevator, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the net force is the difference between the gravitational force acting on the man and the normal force exerted by the scale.
Step 1: Calculate the gravitational force acting on the man.
The gravitational force can be calculated using the formula:
Force = mass * gravity
Given that the mass of the man is 69 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we have:
Gravitational force = 69 kg * 9.8 m/s^2 = 676.2 N
Step 2: Calculate the net force.
The net force can be calculated by subtracting the normal force from the gravitational force:
Net force = Gravitational force - Normal force
Given that the normal force is 823 N, we have:
Net force = 676.2 N - 823 N = -146.8 N
Note: The negative sign indicates that the net force is in the opposite direction to the gravitational force.
Step 3: Calculate the acceleration of the elevator.
Using Newton's second law, we can rearrange the formula to solve for acceleration:
Acceleration = Net force / mass
Plugging in the values, we have:
Acceleration = -146.8 N / 69 kg ≈ -2.13 m/s^2
Therefore, the acceleration of the elevator is approximately 2.13 m/s^2 upwards.