A spring is attached to the ceiling of an elevator and a block of mass M is suspended from the spring. In Case A, the elevator moves up with a constant speed of 5 m/s. In Case B, the elevator moves downward with a constant speed of 10 m/s. Will the string be stretched more in Case A or Case B or is the stretch the same in both cases? Explain.
The string will be stretched more in Case B. This is because when the elevator moves downward, the block of mass M is accelerated downward due to gravity, and the spring must stretch to absorb the additional energy. When the elevator moves upward, the block of mass M is decelerated due to gravity, and the spring must compress to absorb the energy. Therefore, the spring will be stretched more in Case B.
In Case A, when the elevator moves up with a constant speed of 5 m/s, the block of mass M experiences an apparent increase in weight (due to the upward acceleration of the elevator) which is equal to M × g, where g is the acceleration due to gravity. The spring elongates to balance this increase in weight, effectively stretching.
In Case B, when the elevator moves downward with a constant speed of 10 m/s, the block of mass M experiences an apparent decrease in weight (due to the downward acceleration of the elevator) which is equal to M × g. The spring compresses to balance this decrease in weight, effectively stretching.
Since the apparent weight changes in opposite directions in the two cases, the stretching of the spring will be greater in Case B when the elevator moves downward with a constant speed of 10 m/s.
To determine whether the spring will be stretched more in Case A or Case B, we need to consider the forces acting on the block in each situation.
In Case A, when the elevator moves up with a constant speed, the block experiences a force due to its weight pulling it downward and the equal and opposite force from the spring pulling it upward. Since the speed remains constant, the acceleration of the block is zero. Consequently, the net force acting on the block is also zero, meaning the force from the spring is balanced by the force due to gravity. Therefore, the spring will not be stretched in Case A, and the stretch is zero.
In Case B, when the elevator moves downward with a constant speed, the block experiences the same forces as in Case A but in the opposite direction. The weight of the block still pulls it downward, and the spring exerts an upward force. However, since the elevator is moving downward, the acceleration is once again zero. This means the net force on the block is zero, and the force from the spring is again balanced by the force due to gravity. Similar to Case A, the spring will not be stretched in Case B, and the stretch is also zero.
Therefore, the spring will not be stretched more in either Case A or Case B. The stretch of the spring is the same in both cases, and it remains at zero.