a 61.6 kg person on a diet stand on a bathroom scale in an elevator. to their great pleasure the scale reads 46.8 kg. what is the acceleration of the elevator? (up is positive)
To determine the acceleration of the elevator, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F = m * a).
In this case, the force acting on the person is their weight, which can be calculated using their mass and the acceleration due to gravity (9.8 m/s²):
Weight = Mass * Acceleration due to gravity
F = m * g
Where:
Weight = Force (in Newtons)
Mass = 61.6 kg
Acceleration due to gravity (g) = 9.8 m/s²
Weight = 61.6 kg * 9.8 m/s²
Weight = 603.68 N
Now, let's consider the forces acting on the person in the elevator. There are two forces involved:
1. Gravity acting downward, which is the person's weight (603.68 N).
2. The normal force acting upward, exerted by the bathroom scale.
In an elevator, the normal force can be greater or lesser than the weight of the person depending on the direction of acceleration. Since the scale reads 46.8 kg, which is less than the person's actual weight, we can infer that the elevator is accelerating upward.
The net force acting on the person can be calculated by taking the difference between the normal force (upward) and the weight (downward):
Net Force = Normal Force - Weight
Since the scale reading is equal to the normal force, we can rewrite the equation as:
Net Force = 46.8 kg * 9.8 m/s² - 61.6 kg * 9.8 m/s²
Net Force = (46.8 kg - 61.6 kg) * 9.8 m/s²
Net Force = -14.8 kg * 9.8 m/s²
Net Force = -145.04 N
The negative sign indicates that the net force is in the opposite direction of the acceleration, confirming that the elevator is moving upward.
Now, we can calculate the acceleration using Newton's second law:
Net Force = Mass * Acceleration
-145.04 N = 61.6 kg * a
Solving for acceleration:
a = -145.04 N / 61.6 kg
a ≈ -2.35 m/s²
Therefore, the acceleration of the elevator is approximately -2.35 m/s² (negative indicating upward direction).