Question

You are standing on a bathroom scale in an elevator in a tall building. Your mass is 64 kg. The elevator starts from rest and travels upward with a speed that varies with time according to \(v(t) =\) (3.0 m/s\(^2\))\(t\) \(+\) (0.20 m/s\(^3\))\(t^2\). When \(t =\) 4.0 s, what is the reading on the bathroom scale?

Short answer

The bathroom scale reads 921.6 N at \(t = 4.0\) s.

Step-by-step solution

Determine the Acceleration of the Elevator

To find the acceleration of the elevator, we need to differentiate the velocity equation with respect to time, since the derivative of velocity is acceleration. The equation given is \(v(t) = 3.0t + 0.20t^2\). Differentiating it gives: \[ a(t) = \frac{d}{dt}(3.0t + 0.20t^2) = 3.0 + 2(0.20)t = 3.0 + 0.40t \] Substitute \(t = 4.0 \) s into the equation to find the acceleration at that time: \[ a(4.0) = 3.0 + 0.40\times4.0 = 4.6 \text{ m/s}^2 \]

Calculate the Net Force

To find the net force acting on you in the elevator, use Newton's second law, \( F = ma \). Your mass (\( m \)) is 64 kg, and the acceleration (\( a \)) at \(t = 4.0 \) s is 4.6 m/s² from Step 1. The net force is therefore: \[ F_{\text{net}} = m \cdot a = 64 \text{ kg} \times 4.6 \text{ m/s}^2 = 294.4 \text{ N} \]

Consider the Forces on the Bathroom Scale

In an accelerating system like an elevator, the reading on a scale reflects not just your weight but also the net force resulting from acceleration. The total force exerted by the scale equals your apparent weight while accelerating, which is the sum of your true weight and the net force due to acceleration. The force due to gravity (your true weight) is \(mg\): \[ F_{\text{gravity}} = m \cdot g = 64 \text{ kg} \times 9.8 \text{ m/s}^2 = 627.2 \text{ N} \].

Calculate the Scale Reading

For the scale to read your apparent weight, it must exert a force equal to your true weight plus the net force due to elevator's acceleration. That is: \[ F_{\text{scale}} = F_{\text{gravity}} + F_{\text{net}} = 627.2 \text{ N} + 294.4 \text{ N} = 921.6 \text{ N} \] This is the force the bathroom scale measures, equivalent to the apparent weight in newtons.

Key concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. In this exercise, the focus is on the motion of an elevator moving vertically in a building. Its velocity changes over time, and the equation for this velocity is given by: \(v(t) = 3.0t + 0.20t^2\).Differentiating this equation with respect to time determines the acceleration: \(a(t) = \frac{dv}{dt} = 3.0 + 0.40t\).Acceleration exists because the elevator speeds up over time. Understanding this principle is fundamental when discussing more complex concepts like apparent weight and the forces described in the free-body diagram.

• Kinematics ignores forces - it describes motion.
• Velocity is a function of time in this context.
• Acceleration is derived from the velocity equation.

Apparent Weight

When you stand on a scale in a moving elevator, the reading may differ from your actual weight. This reading is called your apparent weight, which varies based on the elevator's acceleration. During acceleration, the forces acting on your body make you feel heavier or lighter. When the elevator accelerates upward, as in this problem, your apparent weight increases. This is because the scale must exert an additional force equal to the net force caused by the elevator accelerating. This force adds to your real weight, making the scale read higher.

• Apparent weight differs from actual weight in an accelerating system.
• Apparent weight increases with upward acceleration.
• The scale reading reflects both real weight and net force due to motion.

Free-Body Diagram

A free-body diagram is a helpful tool in physics for visualizing and analyzing the forces acting on an object. In this exercise, creating a free-body diagram of the person in the elevator would help you see:

• The gravitational force ( \(F_{\text{gravity}} = mg\) ) acting downwards.
• The force due to the elevator's acceleration acting upwards.
• The normal force by the scale acting upwards, equivalent to the apparent weight.

This diagram allows you to understand how these forces interact to produce the apparent weight. Calculating the forces individually using Newton's second law will lead to a correct prediction of what the bathroom scale reads during upward acceleration.The forces' balance shows how Newton's second law operates in dynamic conditions, where multiple forces contribute to the net result seen in scale readings.