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Chapter 8: Problem 52

A large fake cookie sliding on a horizontal surface is attached to one end of a horizontal spring with spring constant \(k=400 \mathrm{~N} / \mathrm{m} ;\) the other end of the spring is fixed in place. The cookie has a kinetic energy of \(20.0 \mathrm{~J}\) as it passes through the spring's equilibrium position. As the cookie slides, a frictional force of magnitude \(10.0 \mathrm{~N}\) acts on it. (a) How far will the cookie slide from the equilibrium position before coming momentarily to rest? (b) What will be the kinetic energy of the cookie as it slides back through the equilibrium position?

Short Answer

Expert verified
(a) The cookie slides 0.3 meters before stopping. (b) The cookie's kinetic energy is 14 J when passing through equilibrium again.

Step by step solution

01

Understand the Problem

A cookie is attached to a spring and is sliding on a surface with friction. Given the spring constant (400 N/m), initial kinetic energy (20 J), and friction force (10 N), we need to determine how far it slides before stopping and its kinetic energy when it returns to the starting point.
02

Set up the Energy Conservation Equation

Initially, the mechanical energy of the system is composed of kinetic energy only: 20 J. As the cookie moves, it loses energy due to the work done against friction. The equation is: \( K + U - W_{friction} = K' + U' \), where \(K = 20 \text{ J}\) (initial kinetic energy), \(U = 0\) (initial potential energy at equilibrium), \(W_{friction} = f \cdot d\), and the final kinetic energy \(K' = 0\) at the point where the cookie stops.
03

Calculate Work Done by Friction

The work done by friction is calculated using \(W_{friction} = f \cdot d\), where \(f = 10 \text{ N}\). This work converts the initial kinetic energy into spring potential energy until the cookie stops.
04

Determine the Spring Potential Energy at Maximum Displacement

The spring potential energy at maximum displacement is \(U' = \frac{1}{2}kx^2\). Since all initial kinetic energy is converted to spring potential energy and work done by friction: \(K - W_{friction} = U'\), thus \(20 - 10x = \frac{1}{2} \cdot 400 \cdot x^2\).
05

Solve for Maximum Displacement

Rearrange the equation to solve for \(x\): \(0 = 200x^2 + 10x - 20\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 200, b = 10, c = -20\). Solving gives us \(x = 0.3\) meters after taking the positive root.
06

Analyze Energy when Returning to Equilibrium

When the cookie returns to the equilibrium position, all spring potential energy plus the energy lost to friction converts back to kinetic energy. Calculate: \(K' = U' - W_{friction}\) when \(x = 0\). Given already calculated work \(W_{friction} = 10 \cdot 0.6 = 6 \text{ J}\), we find \(K' = 20 - 6 = 14 \text{ J}\).

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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 a fundamental concept in physics that describes the energy an object possesses due to its motion. It is dependent on both the mass and the velocity of the object. Mathematically, it is expressed as: \[ KE = \frac{1}{2}mv^2 \]where
  • \( KE \) is the kinetic energy,
  • \( m \) is the mass, and
  • \( v \) is the velocity of the object.
For a larger mass or a higher velocity, the kinetic energy increases. In our exercise, the cookie initially has a kinetic energy of 20.0 J as it slides through the equilibrium position of the spring. This energy is what allows the cookie to continue moving against forces like friction.
Spring Potential Energy
Spring potential energy is stored in compressed or stretched springs. It represents the energy stored when an object is displaced from its equilibrium position. The equation governing this is:\[ U_s = \frac{1}{2}kx^2 \]where
  • \( U_s \) is the spring potential energy,
  • \( k \) is the spring constant, and
  • \( x \) is the displacement from the equilibrium position.
In our problem, as the cookie moves, it compresses the spring, transferring its kinetic energy into spring potential energy until it momentarily stops. The spring constant \( k = 400 \, \text{N/m} \) tells us how stiff the spring is, and it is crucial in determining how much energy is stored for a given displacement.
Frictional Force
Frictional force is the resistive force that opposes motion when two surfaces are in contact. It causes energy to be transferred from kinetic energy into heat or other forms, and it is given by:\[ F_f = \mu N \]where
  • \( F_f \) is the frictional force,
  • \( \mu \) is the coefficient of friction, and
  • \( N \) is the normal force.
In our exercise, instead of a known \( \mu \), we directly work with the frictional force magnitude of 10.0 N. This force does work against the motion of the cookie, affecting its energy conservation by converting part of that energy into non-mechanical forms such as heat.
Work-Energy Principle
The work-energy principle is a vital concept in physics that connects the work done on an object to its change in energy. Mathematically, it is expressed as:\[ W = \Delta KE + \Delta U \]where
  • \( W \) is the work done,
  • \( \Delta KE \) is the change in kinetic energy, and
  • \( \Delta U \) is the change in potential energy.
In the given problem, the work done by friction as the cookie moves is calculated using \( W_{friction} = f \cdot d \), where \( d \) is the distance moved. This work reduces the cookie's kinetic energy and converts it into spring potential energy when the cookie compresses the spring and stops temporarily. Understanding the work-energy principle helps to comprehend how mechanical energy transitions between different forms during an object's motion.

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Most popular questions from this chapter

A certain spring is found not to conform to Hooke's law. The force (in newtons) it exerts when stretched a distance \(x\) (in meters) is found to have magnitude \(52.8 x+38.4 x^{2}\) in the direction opposing the stretch. (a) Compute the work required to stretch the spring from \(x=0.500 \mathrm{~m}\) to \(x=1.00 \mathrm{~m} .(\mathrm{b})\) With one end of the spring fixed, a particle of mass \(2.17 \mathrm{~kg}\) is attached to the other end of the spring when it is stretched by an amount \(x=1.00 \mathrm{~m}\). If the particle is then released from rest, what is its speed at the instant the stretch in the spring is \(x=0.500 \mathrm{~m} ?(\mathrm{c})\) Is the force exerted by the spring conservative or nonconservative? Explain.

A metal tool is sharpened by being held against the rim of a wheel on a grinding machine by a force of \(180 \mathrm{~N}\). The frictional forces between the rim and the tool grind off small pieces of the tool. The wheel has a radius of \(20.0 \mathrm{~cm}\) and rotates at \(2.50 \mathrm{rev} / \mathrm{s}\). The coefficient of kinetic friction between the wheel and the tool is \(0.320 .\) At what rate is energy being transferred from the motor driving the wheel to the thermal energy of the wheel and tool and to the kinetic energy of the material thrown from the tool?

A constant horizontal force moves a \(50 \mathrm{~kg}\) trunk \(6.0 \mathrm{~m}\) up a \(30^{\circ}\) incline at constant speed. The coefficient of kinetic friction is \(0.20\). What are (a) the work done by the applied force and (b) the increase in the thermal energy of the trunk and incline?

A \(50 \mathrm{~g}\) ball is thrown from a window with an initial velocity of \(8.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(30^{\circ}\) above the horizontal. Using energy methods, determine (a) the kinetic energy of the ball at the top of its flight and \((\mathrm{b})\) its speed when it is \(3.0 \mathrm{~m}\) below the window. Does the answer to (b) depend on (c) the mass of the ball or (d) the initial angle?

Approximately \(5.5 \times 10^{6} \mathrm{~kg}\) of water falls \(50 \mathrm{~m}\) over Niagara Falls each second. (a) What is the decrease in the gravitational potential energy of the water-Earth system each second? (b) If all this energy could be converted to electrical energy (it cannot be), at what rate would electrical energy be supplied? (The mass of \(1 \mathrm{~m}^{3}\) of water is \(\left.1000 \mathrm{~kg} .\right)(\mathrm{c})\) If the electrical energy were sold at 1 cent \(/ \mathrm{kW} \cdot \mathrm{h}\), what would be the yearly income?
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