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Chapter 37: Problem 37

(a) By what percentage does your rest mass increase when you climb 30 \(\mathrm{m}\) to the top of a ten-story building? Are you aware of this increase? Explain. (b) By how many grams does the mass of a \(12.0-\mathrm{g}\) spring with force constant 200 \(\mathrm{N} / \mathrm{cm}\) change when you compress it by 6.0 \(\mathrm{cm} ?\) Does the mass increase or decrease? Would you notice the change in mass if you were holding the spring? Explain.

Short Answer

Expert verified
(a) Mass increases by ~3.29e-16%; not noticeable. (b) Mass increases by ~4.00e-13g; not noticeable.

Step by step solution

01

Understanding the Problem

We need to solve two questions regarding mass change due to energy change. The first part involves calculating the change in rest mass when moving to a higher gravitational potential. The second part involves determining the change in mass of a compressed spring using the energy stored by virtue of its compression.
02

Calculate Increase in Potential Energy from Climbing

When you climb a height of 30 meters, the potential energy increases by \[ \Delta U = mgh \]where \( m \) is the mass of the person, \( g = 9.81 \, \text{m/s}^2 \), and \( h = 30 \, \text{m} \). Assume a person's rest mass is \( m = 70 \, \text{kg} \). The increase in potential energy is then \[ \Delta U = 70 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 30 \, \text{m} \approx 20655 \, \text{J} \].
03

Relate Energy Change to Mass Change

Using Einstein's mass-energy equivalence principle, the change in mass \( \Delta m \) due to energy change \( \Delta U \) is given by: \[ \Delta m = \frac{\Delta U}{c^2} \]where \( c = 3.00 \times 10^8 \, \text{m/s} \). Substitute \( \Delta U = 20655 \, \text{J} \): \[ \Delta m = \frac{20655}{(3.00 \times 10^8)^2} \approx 2.3 \times 10^{-13} \, \text{kg} \].
04

Calculate Percentage Mass Increase

The percentage increase in mass is calculated by: \[ \text{Percentage} = \left( \frac{\Delta m}{m} \right) \times 100 \% \approx \left( \frac{2.3 \times 10^{-13} \, \text{kg}}{70 \, \text{kg}} \right) \times 100\% \approx 3.29 \times 10^{-16} \% \].
05

Evaluate Perception of Mass Change

The calculated percentage increase in mass \( \approx 3.29 \times 10^{-16} \% \) is incredibly small. Humans cannot perceive such a minuscule change in mass, so it is unlikely that you would notice it while climbing.
06

Calculate Energy Stored in Compressed Spring

The potential energy stored in a compressed spring is given by: \[ \Delta U = \frac{1}{2} k x^2 \]where \( k = 200 \, \text{N/cm} \) and \( x = 6.0 \, \text{cm} \). Convert \( k \) to \( \text{N/m} \) and substitute: \[ \Delta U = \frac{1}{2} \times 20000 \, \text{N/m} \times (0.06 \, \text{m})^2 = 36 \, \text{J} \].
07

Calculate Mass Change of Spring

Use the mass-energy equivalence again: \[ \Delta m = \frac{\Delta U}{c^2} = \frac{36 \, \text{J}}{(3.00 \times 10^8 \, \text{m/s})^2} \approx 4.00 \times 10^{-16} \, \text{kg} \].
08

Evaluate Perception of Spring's Mass Change

Convert kilogram change to grams: \(4.00 \times 10^{-16} \text{kg} \approx 4.00 \times 10^{-13} \text{g} \).This change in mass is much smaller than what can be detected by human senses, so you would not perceive any difference in weight when holding the spring after compression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy
Potential energy is a form of energy that is stored within an object due to its position relative to some zero position. A common type of potential energy is gravitational potential energy, which we will discuss further in this article.

For example, when you lift an object, you increase its potential energy because you're doing work against the gravitational force. This increase in potential energy can be calculated using the formula:
  • \( \Delta U = mgh \)
  • where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point.
Potential energy is crucial in understanding various physical systems and is often transformed into kinetic energy or other forms of energy when released.
Mass Change
Mass change is an interesting phenomenon that comes into play due to Einstein's mass-energy equivalence principle. This principle is summarized by the famous equation:
  • \( E = mc^2 \)
  • where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light in a vacuum.
In a physical system, when energy is added or removed, the mass of the system changes, albeit usually by a very tiny amount.

For example, climbing stairs or compressing a spring leads to a change in potential energy, which can lead to a very slight change in mass. While these changes in mass are typically too small to detect in everyday scenarios, they illustrate the deep connection between mass and energy.
Compressed Spring
When a spring is compressed, it stores potential energy as a result of its deformation. This energy is referred to as elastic potential energy, and it can be calculated using the formula:
  • \( \Delta U = \frac{1}{2} k x^2 \)
  • where \( k \) is the spring constant, measuring the stiffness of the spring, and \( x \) is the amount of compression from its equilibrium position.

The relationship between the potential energy stored in a compressed spring and mass change can be understood through the mass-energy equivalence principle.

While the change in mass due to compressing a spring is typically minute and imperceptible, this concept showcases how energy changes within an internal system can affect mass.
Gravitational Potential Energy
Gravitational potential energy is the energy an object possesses because of its position in a gravitational field. This is one of the most commonly encountered forms of potential energy.

When an object, such as a person, is raised to a height, it gains gravitational potential energy. It can be computed using:
  • \( U = mgh \)
  • where \( m \) is mass, \( g \) is gravitational acceleration, and \( h \) is the height above a reference point.
As we climb up or down, this energy changes, yet the resulting mass change remains virtually invisible because the numbers involved are very small.

Thus, even though our mass technically increases due to climbing a building, it's too slight for us to notice any difference without precise instruments.

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Most popular questions from this chapter

The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed \(V\) relative to the lab frame is $$v=\frac{c}{n}+k V$$ where \(n=1.333\) is the index of refraction of water. Fizeau called \(k\) the draging coefficient and obtained an experimental value of \(k=0.44 .\) What value of \(k\) do you calculate from relativistic transformations?

After being produced in a collision between elementary particles, a positive pion \(\left(\pi^{+}\right)\) must travel down a 1.90 -km-long tube to reach an experimental area. A \(\pi^{+}\) particle has an average lifetime (measured in its rest frame) of \(2.60 \times 10^{-8} \mathrm{s}\) ; the \(\pi^{+}\) we are considering has this lifetime. How fast must the \(\pi^{+}\) travel if it is not to decay before it reaches the end of the tube? (Since \(u\) will be very close to \(c,\) write \(u=(1-\Delta) c\) and give your answer in terms of \(\Delta\) rather than \(u .\) (b) The \(\pi^{+}\) has a rest energy of 139.6 \(\mathrm{MeV} .\) What is the total energy of the \(\pi^{+}\) at the speed calculated in part (a)?

A particle with mass \(m\) accelerated from rest by a constant force \(F\) will, according to Newtonian mechanics, continue to accelerate without bound; that is, as \(t \rightarrow \infty, v \rightarrow \infty .\)Show that according to relativistic mechanics, the particle's speed approaches \(c\) as \(t \rightarrow \infty\) [Note: A useful integral is \(\int\left(1-x^{2}\right)^{-3 / 2} d x=\) \(x / \sqrt{1-x^{2}}\)]

Compute the kinetic energy of a proton (mass \(1.67 \times\) \(10^{-27} \mathrm{kg}\) ) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic for speeds of (a) \(8.00 \times 10^{7} \mathrm{m} / \mathrm{s}\) and (b) \(2.85 \times\) \(10^{8} \mathrm{m} / \mathrm{s} .\)

The positive muon \(\left(\mu^{+}\right),\) an unstable particle, lives on average \(2.20 \times 10^{-6} \mathrm{s}\) (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of \(0.900 c,\) what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?
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