/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 You and your bicycle have combin... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 73

You and your bicycle have combined mass 80.0 \(\mathrm{kg}\) . When you reach the base of a bridge, you are traveling along the road at 5.00 \(\mathrm{m} / \mathrm{s}(\mathrm{Fig} .6 .35) .\) At the top of the bridge, you have climbed a vertical distance of 5.20 \(\mathrm{m}\) and have slowed to 1.50 \(\mathrm{m} / \mathrm{s}\) . You can ignore work done by friction and any inefficiency in the bike or your legs. (a) What is the total work done on you and your bicycle when you go from the base to the top of the bridge? (b) How much work have you done with the force you apply to the pedals?

Short Answer

Expert verified
Total work done: 3160.4 J. Work done by you: 3160.4 J.

Step by step solution

01

Calculate Initial and Final Kinetic Energy

The initial kinetic energy \( KE_i \) can be calculated using the formula \( KE_i = \frac{1}{2} m v_i^2 \), where \( m = 80.0 \, \mathrm{kg} \) and \( v_i = 5.00 \, \mathrm{m/s} \). Plug in the values: \[ KE_i = \frac{1}{2} \times 80.0 \, \mathrm{kg} \times (5.00 \, \mathrm{m/s})^2 = 1000 \, \mathrm{J} \].The final kinetic energy \( KE_f \) is given by \( KE_f = \frac{1}{2} m v_f^2 \), where \( v_f = 1.50 \, \mathrm{m/s} \). Plug in the values:\[ KE_f = \frac{1}{2} \times 80.0 \, \mathrm{kg} \times (1.50 \, \mathrm{m/s})^2 = 90 \, \mathrm{J} \].
02

Calculate Change in Potential Energy

The change in potential energy \( \Delta PE \) is determined using the formula \( \Delta PE = m g h \), where \( g = 9.81 \, \mathrm{m/s^2} \) is the acceleration due to gravity and \( h = 5.20 \, \mathrm{m} \) is the change in height.\[ \Delta PE = 80.0 \, \mathrm{kg} \times 9.81 \, \mathrm{m/s^2} \times 5.20 \, \mathrm{m} = 4070.4 \, \mathrm{J} \].
03

Calculate Total Work Done

The total work done \( W_{\text{total}} \) is the change in mechanical energy, which is given by \( W_{\text{total}} = (KE_f - KE_i) + \Delta PE \).Substitute the values:\[ W_{\text{total}} = (90 \, \mathrm{J} - 1000 \, \mathrm{J}) + 4070.4 \, \mathrm{J} = 3160.4 \, \mathrm{J} \].
04

Calculate Work Done by Pedal Force

Since friction is negligible and no energy is lost to inefficiencies, the work done by you with the pedal force is equal to the total work done on the system.Thus, the work done by the pedal force is \( 3160.4 \, \mathrm{J} \).

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Key Concepts

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

Kinetic Energy
Kinetic energy is all about motion. Whenever an object is moving, it possesses kinetic energy. The faster the object moves, the more kinetic energy it has. This energy depends on two main factors:
  • Mass (\( m \)) - Simply put, the heavier the object, the more energy it can carry in motion.
  • Velocity (\( v \)) - The speed of the object plays an even bigger role; the kinetic energy actually increases with the square of the velocity.
The kinetic energy (\( KE \)) of an object is calculated using the formula:\[ KE = \frac{1}{2} mv^2 \]In the exercise, you started cycling at 5.00 m/s reaching a kinetic energy of 1000 J. By the time you reached the top of the bridge, your speed reduced to 1.50 m/s, cutting down the kinetic energy to just 90 J. This reduction in kinetic energy is one key part of understanding the work-energy principle.
Potential Energy
Potential energy is stored energy based on an object's position or condition. For height changes, this type of energy is often called gravitational potential energy. It depends on:
  • Mass (\( m \)) of the object - Again, heavier objects have more potential for stored energy.
  • Height (\( h \)) - The higher you are, the more potential energy you acquire, due to Earth's gravity.
  • Acceleration due to gravity (\( g \)) - On Earth, this value is approximately 9.81 m/s虏.
The formula for gravitational potential energy (\( PE \)) is:\[ PE = mgh \]In our scenario, by ascending 5.20 meters, you gained 4070.4 J of potential energy. This energy was transformed from the work you did as you pedaled up the bridge! The increase in potential energy is crucial in figuring out how much work you did.
Mechanical Energy
Mechanical energy is the sum of both kinetic and potential energies in a system. It's essentially the total energy from motion and position combined. In this exercise, both kinetic and potential energies are at play:
  • At the beginning at the base of the bridge, you mostly have kinetic energy (1000 J).
  • By the top of the bridge your potential energy increased as your position changed to a higher elevation.
The total mechanical energy at any point is given by:\[ ME = KE + PE \]The work-energy principle states that the total work done on an object is equal to its change in mechanical energy. As you cycled up, you lost some kinetic energy while gaining potential energy. The net work done was 3160.4 J, matching the increase in your mechanical energy. This tells us about how much energy you exerted using your pedal force to get to the top of the bridge. Understanding these energy transformations can enrich your grasp of not just physics problems but also real-world energy scenarios!

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Most popular questions from this chapter

A pump is required to lift 800 \(\mathrm{kg}\) of water (about 210 gallons) per minute from a well 14.0 \(\mathrm{m}\) deep and eject it with a speed of 18.0 \(\mathrm{m} / \mathrm{s}\) , (a) How much work is done per minute in lifting the water? (b) How much work is done in giving the water the kinetic energy it has when ejected? (c) What must be the power output of the pump?

A moving electron has kinetic energy \(K_{1}\) . After a net amount of work \(W\) has been done on it, the electron is moving one-quarter as fast in the opposite direction. (a) Find \(W\) in terms of \(K_{1}\) . (b) Does your answer depend on the final direction of the electron's motion?

Working Like a Horse. Your job is to lift \(30-k g\) crates a vertical distance of 0.90 \(\mathrm{m}\) from the ground onto the bed of a truck. (a) How many crates would you have to load onto the truck in 1 minute for the average power output you use to lift the crates to equal 0.50 \(\mathrm{hp} ?\) (b) How many crates for an average power output of 100 \(\mathrm{W} ?\)

Springs in Parallel. Two springs are in parallel if they are parallel to each other and are connected at their ends (Figure 6.33\()\) . We can think of this combination as being equivalent to a single spring. The force constant of the equivalent single spring is called the effective forceconstant, \(k_{\text { eff }},\) of the combination. (a) Show that the effective force constant of this combination is \(k_{\mathrm{eff}}=k_{1}+k_{2} .\) (b) Generalize this result for \(N\) springs in parallel.

The Grand Coulee Dam is 1270 \(\mathrm{m}\) long and 170 \(\mathrm{m}\) high. The electrical power output from generators at its base is approximately 2000 MW. How many cubic meters of water must flow from the top of the dam per second to produce this amount of power if 92\(\%\) of the work done on the water by gravity is converted to electrical energy? (Each cubic meter of water has a mass of 1000 \(\mathrm{kg}\) .)
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