Problem 39 - College Physics

Question

The weight of a 500 -kg object is 4900 N. (a) When the object is on a moving elevator, its measured weight could be (1) zero, (2) between zero and \(4900 \mathrm{~N}\), (3) more than \(4900 \mathrm{~N},\) (4) all of the preceding. Why? (b) Describe the motion if the object's measured weight is only \(4000 \mathrm{~N}\) in a moving elevator.

Step-by-Step Solution

Answer

(a) The apparent weight can be all options based on elevator motion. (b) The elevator is accelerating downwards at 1.8 m/s².

Step 1: Understanding Weight on an Elevator

The weight of an object in an elevator can change due to the acceleration of the elevator. This is because the apparent weight is the normal force exerted by the surface, which changes with acceleration. If the elevator accelerates upwards, the apparent weight increases; if it accelerates downwards, the apparent weight decreases.

Step 2: Analyzing Possible Weight Readings

(a) The measured weight can be: (1) zero if the elevator is in free fall, (2) between zero and 4900 N if the elevator is accelerating downwards but not in free fall, (3) more than 4900 N if the elevator is accelerating upwards, and (4) all of the preceding situations are possible depending on the elevator's motion.

Step 3: Calculating the Apparent Weight for 4000 N

(b) The true weight of the object is 4900 N. If the apparent weight is 4000 N, it means the elevator is accelerating downwards. We can find the acceleration using the equation: \( F_{apparent} = mg - ma \), where \( F_{apparent} = 4000 \mathrm{~N}, m = 500 \mathrm{~kg}, \text{and} \; g = 9.8 \mathrm{~m/s}^2 \).

Step 4: Solving for Acceleration

Rearrange the equation to solve for acceleration \( a \): \( 4000 = 4900 - 500a \). Simplify and solve for \( a \): \[ 500a = 4900 - 4000 \]\[ 500a = 900 \]\[ a = \frac{900}{500} = 1.8 \mathrm{~m/s}^2 \]. So, the elevator is accelerating downwards at \(1.8 \mathrm{~m/s}^2 \).

Key Concepts

Apparent Weight

Apparent weight describes how heavy an object feels when it is in a non-inertial frame of reference, like an elevator that's either speeding up or slowing down. An object in an elevator experiences changes in normal force, which alter its apparent weight. This force adjustment can make you feel heavier or lighter compared to when you are standing still on the ground. The key point to understand is that while your mass stays constant, the force you feel as weight can change due to acceleration.
This happens because apparent weight is the force exerted by the floor of the elevator, changing with motion:

• If the elevator accelerates upwards, the floor pushes more strongly against you, increasing your apparent weight.
• If it accelerates downwards, the push is weaker, decreasing your apparent weight.
• If it falls freely ( No normal force at all!), your apparent weight is zero.

This concept can help us understand the movement of any object in varying conditions, like an astronaut feeling weightlessness in orbit.

Acceleration

Acceleration is the rate at which the velocity of an object changes over time. In the context of an elevator, acceleration explains why your apparent weight can fluctuate. When an elevator changes speed or direction, it exerts additional forces on you, altering how heavy you feel.
Let's break it down with some examples:

• When an elevator starts moving upward or slows down while going down, it accelerates positively. In this scenario, you feel heavier because the floor pushes harder against you.
• Conversely, if the elevator starts moving downward or slows down while going up, it accelerates negatively, making you feel lighter as the push from the floor decreases.
• If it descends quickly enough, it might feel like you're weightless, similar to the sensation experienced during a free fall.

Understanding the effects of acceleration helps in quantifying how motion affects apparent weight using formulas like: \[ F_{apparent} = mg - ma \]where \( m \) is mass, \( g \) is gravitational acceleration, and \( a \) is the elevator's acceleration.

Newton's Laws

Newton's Laws provide the foundation for understanding motion and forces experienced in scenarios like riding an elevator. The pivotal laws applicable here are the Second and Third Laws:

• Newton's Second Law states that the acceleration of an object is dependent on the net forces acting on it and is expressed as \( F = ma \). This is crucial in calculating the apparent weight changes: as the elevator accelerates, the net force changes, reflecting in your sensation of weight.
• Newton's Third Law tells us that for every action, there is an equal and opposite reaction. When you stand in an accelerating elevator, the force you exert on the floor (your weight) results in an equal and opposite force (normal force) acting on you. This is your apparent weight, and it varies according to the elevator's motion.

Newton's Laws show us that we aren't lighter or heavier because our mass changes. Instead, it's because the forces at play (like tension and normal force) shift as per our movement in different frames, giving us the sensation of differing weights.