Question

An object is hung from a spring scale attached to the ceiling of an elevator. The scale reads \(65 \mathrm{~N}\) when the elevator is standing still. ( \(a\) ) What is the reading when the elevator is moving upward with a constant speed of \(7.6 \mathrm{~m} / \mathrm{s} ?(b)\) What is the reading of the scale when the elevator is moving upward with a speed of \(7.6 \mathrm{~m} / \mathrm{s}\) and decelerating at \(2.4 \mathrm{~m} / \mathrm{s}^{2} ?\)

Short answer

The reading of the spring when the elevator is moving upward with a constant speed is \(65 \mathrm{~N}\). When the elevator is decelerating at \(2.4 \mathrm{~m/s^{2}}\), the reading is \(49.1 \mathrm{~N}\).

Step-by-step solution

Calculate Mass of the Object

When the elevator is stationary, the scale reading shows the weight of the object. This is given by the formula Weight = Mass x Gravitational Force. From the exercise, we know that the gravitational force is \(9.8 \mathrm{~m/s^{2}}\) and the weight is \(65 \mathrm{~N}\). We rearrange the formula and find the mass: Mass = Weight / Gravitational Force, Mass = \(65 \mathrm{~N} / 9.8 \mathrm{~m/s^{2}} = 6.63 \mathrm{~kg}\).

Determine the Reading at Constant Speed

When the elevator is moving at a constant speed, the net force on the object is zero because there is no acceleration (Newton’s first law of motion). Therefore, the scale reading remains the same as when the elevator is at rest, which is \(65 \mathrm{~N}\).

Determine the Reading during Deceleration

When the elevator is moving upward and decelerating, the net force on the object is given by the equation Net Force = Mass x (Gravitational Force - Deceleration). Substituting the known values into the formula: Net Force = \(6.63 \mathrm{~kg} * (9.8 \mathrm{~m/s^{2}} - 2.4 \mathrm{~m/s^{2}}) = 49.1 \mathrm{~N}\). This is the reading of the scale during deceleration.

Key concepts

Newton's laws of motion

Understanding Newton's laws of motion is fundamental when addressing an elevator physics problem. Sir Isaac Newton's first law, often called the law of inertia, tells us that an object will remain at rest or move at a constant speed in a straight line unless acted upon by a net force. In the context of the elevator problem, when the elevator travels upwards with constant velocity, there is no net force acting on the object, meaning that the forces are balanced. The spring scale will read the same as when the elevator is at rest because the object's velocity does not influence its weight. This concept is critical for determining the readings of the scale under different conditions of motion.

Gravitational force

Gravitational force is the attraction between two objects with mass. On Earth, this force gives us weight and is directed towards the center of the planet. Its standard value, referred to as 'g', is approximately 9.8 meters per second squared (\(9.8 \text{ m/s}^2\)). For any object, the gravitational force can be calculated by multiplying its mass by 'g'. This provides the basis for understanding the spring scale reading when the elevator is stationary; the scale measures the gravitational force exerted by the object's mass.

Deceleration

Deceleration is the reduction in the speed of an object, or negative acceleration. In our elevator scenario, when the elevator is decelerating, the gravitational force still acts downwards, but now there's an additional force in the opposite direction, slowing the object's ascent. This reduction in the object's upward speed results in a decrease in the spring scale reading relative to when the elevator is at rest. Understanding deceleration is key to analyzing situations where the elevator's speed changes.

Spring scale reading

A spring scale measures the force exerted on it, which can be due to the object's weight or additional forces applied, such as tension or the forces during acceleration or deceleration. The original scale reading when at rest is solely due to gravitational force, but this can change if other force vectors, like deceleration, are introduced. The scale's ability to reflect different forces makes it a practical tool in physics experiments and problems.

Constant speed

An object moving at a constant speed is in a state of dynamic equilibrium, where all forces are balanced, and there is no net force acting on the object (Newton's first law). This means in our elevator example, when it's moving at the constant speed of 7.6 meters per second, the object inside the elevator is not accelerating, and therefore, the spring scale reading will be the same as when the elevator is in a stationary position.

Net force

The net force is the vector sum of all the forces acting on an object. When an object is in equilibrium (either at rest or moving with constant speed), the net force is zero. In the elevator problem, we see that the net force changes when the elevator starts decelerating. To find the new spring scale reading during deceleration, one must consider both gravitational force and the force attributed to deceleration, which in this case is subtracted from the gravitational force because it acts in the opposite direction to it. The net force equation includes both these forces and once calculated, this provides the scale’s reading during deceleration.