A 78-kg man stands on a spring scale in an elevator. Starting from rest, the elevator ascends, attaining its maximum speed of 1.2 m/s in 0.74 s. The elevator travels with this constant speed for 5.0 s, undergoes a uniform negative acceleration for 1.8 s, and then comes to rest. What does the spring scale register in each of the following time intervals?
(a) before the elevator starts to move
(b) during the first 0.74 s of the elevator's ascent
(c) while the elevator is traveling at constant speed
(d) during the elevator's negative acceleration
scalereading=mg+ma where a is upward direction, if downward, negative. a is zero during constant velocity.
To find out what the spring scale registers in each time interval, we need to consider the forces acting on the man and apply Newton's second law of motion. Let's break down the problem into different time intervals:
(a) Before the elevator starts to move:
When the elevator is at rest, there is no net force acting on the man. Therefore, the spring scale will register the man's weight, which is equal to his mass multiplied by the acceleration due to gravity (9.8 m/s^2).
Spring scale reading = mg = 78 kg * 9.8 m/s^2 = 764.4 N
(b) During the first 0.74 s of the elevator's ascent:
Since the elevator is accelerating upwards, there is an additional force acting on the man, the upward force exerted by the elevator. The total force acting on the man is the sum of his weight and the elevator force.
Acceleration = Change in velocity / Time = (1.2 m/s - 0 m/s) / 0.74 s = 1.62 m/s^2
Net force on the man = mass * acceleration = 78 kg * 1.62 m/s^2 = 126.36 N
However, the spring scale registers the normal force exerted by the man on the elevator, which is equal to the magnitude of the net force.
Spring scale reading = 126.36 N
(c) While the elevator is traveling at constant speed:
When the elevator is moving at a constant speed, there is no net force acting on the man. The net force is zero because the man's weight is balanced by the normal force exerted by the elevator.
Spring scale reading = mg = 78 kg * 9.8 m/s^2 = 764.4 N
(d) During the elevator's negative acceleration:
When the elevator undergoes a uniform negative acceleration, the net force on the man is the difference between his weight and the downward force exerted by the elevator.
Acceleration = Change in velocity / Time = (0 m/s - 1.2 m/s) / 1.8 s = -0.67 m/s^2
Net force on the man = mass * acceleration = 78 kg * (-0.67 m/s^2) = -52.26 N
The spring scale reading is the normal force exerted by the man on the elevator, which is equal to the magnitude of the net force.
Spring scale reading = 52.26 N (pointing downwards)
To determine what the spring scale registers in each time interval, we need to analyze the forces acting on the man in the elevator during each period.
(a) Before the elevator starts to move:
When the elevator is at rest, there is no acceleration, and therefore, no net force acting on the man. The only force he experiences is the force due to gravity, which is equal to mg, where m is the mass of the man and g is the acceleration due to gravity. The spring scale will register the force due to gravity, which is the man's weight. Thus, the spring scale will register the man's weight, which can be calculated as:
Weight = mass × acceleration due to gravity = 78 kg × 9.8 m/s^2.
(b) During the first 0.74 s of the elevator's ascent:
In this interval, the elevator is accelerating, but the man is still on the floor of the elevator and not in contact with the spring scale. Therefore, the spring scale will continue to register only the man's weight, as in part (a).
(c) While the elevator is traveling at constant speed:
When the elevator is moving at a constant speed, there is no acceleration. Because of this, the man does not experience any additional forces beyond his weight due to gravity. Therefore, the spring scale will continue to register the man's weight, as in part (a).
(d) During the elevator's negative acceleration:
During the negative acceleration, the elevator is slowing down. Due to this deceleration, there is an additional force acting on the man, directed opposite to the direction of the acceleration. This force is equal to the mass of the man multiplied by the magnitude of the acceleration. The spring scale will register the net force acting on the man, which is the difference between the force due to gravity and the additional force. The additional force can be calculated using Newton's second law, F = ma, where F is the additional force, m is the mass of the man, and a is the acceleration. The spring scale will register the net force, which can be calculated as:
Net Force = Weight - Additional Force = Weight - (mass × acceleration).
To find the additional force, we need to find the acceleration of the elevator during the negative acceleration phase. To do this, we need to know the final velocity of the elevator at the end of the constant speed phase, which is 1.2 m/s. Given that the elevator undergoes a uniform negative acceleration during the next 1.8 seconds until it comes to rest, we can use the formula:
acceleration = (final velocity - initial velocity) / time.
Using this formula, the additional force and the net force can be calculated.
It is important to note that the spring scale will only register the magnitude of the net force acting on the man, and not the direction. If the net force is negative, indicating a force opposite to the direction of gravity, the spring scale will register a lower value than the man's weight. If the net force is positive, the spring scale will register a higher value than the man's weight.