/*! 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 43 A block with mass 0.50 \(\mathrm... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 43

A block with mass 0.50 \(\mathrm{kg}\) is forced against a horizontal spring of negligible mass, compressing the spring a distance of 0.20 \(\mathrm{m}(\) Fig. \(\mathrm{P} 7.43) .\) When released, the block moves on a horizon tal tabletop for 1.00 \(\mathrm{m}\) before coming to rest. The spring constant \(k\) is 100 \(\mathrm{N} / \mathrm{m}\) . What is the coefficient of kinetic friction \(\mu_{\mathrm{k}}\) between the block and the tabletop?

Short Answer

Expert verified
The coefficient of kinetic friction is approximately 0.41.

Step by step solution

01

Determine the Spring Potential Energy

First, calculate the potential energy stored in the spring. The formula for the potential energy stored in a spring is given by \[ U = \frac{1}{2} k x^2 \]where \( k = 100 \, \text{N/m} \) is the spring constant and \( x = 0.20 \, \text{m} \) is the compression distance. Plugging the values in:\[ U = \frac{1}{2} \times 100 \times (0.20)^2 = 2 \, \text{J} \]
02

Relate Work Done by Friction to Energy

Since the block comes to a stop after a 1.00 \( \text{m} \) movement, this energy stored as spring potential energy is transferred to work done against friction. The work done on the block by the friction force is equal to the friction force (\( f_k = \mu_k \cdot F_N \)) times the distance (\( d \)). Hence:\[ \text{Work by friction} = f_k \cdot d \]Since the block comes to rest, the work by friction equals the initial potential energy:\[ \mu_k \cdot m \cdot g \cdot d = 2 \]
03

Solve for Coefficient of Kinetic Friction

Rearrange the equation to solve for the coefficient of kinetic friction \( \mu_k \):\[ \mu_k = \frac{2}{m \cdot g \cdot d} \]Given:- \( m = 0.50 \, \text{kg} \)- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity)- \( d = 1.00 \, \text{m} \)Plug these values to find \( \mu_k \):\[ \mu_k = \frac{2}{0.50 \times 9.81 \times 1.00} \approx 0.41 \]
04

Finalize the Solution

Confirm that all calculations are correct and consistent with the physical principles involved. The coefficient of kinetic friction \( \mu_k \) is a dimensionless number representing the ratio of the frictional force to the normal force.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Spring Potential Energy
Spring potential energy is the energy stored in a spring when it is compressed or stretched. It's a form of mechanical energy and can be calculated using Hooke's Law.
The formula for spring potential energy is given by:
\[ U = \frac{1}{2} k x^2 \]where:
  • \( U \) is the potential energy in joules (J)
  • \( k \) is the spring constant in newtons per meter (N/m) - it measures the stiffness of the spring
  • \( x \) is the displacement from the spring's equilibrium position in meters (m)
In our example, a spring with a constant \( k = 100 \, \text{N/m} \) is compressed by \( 0.20 \, \text{m} \).
Using these parameters, the stored energy is calculated to be 2 J.
This energy is then available to do work, which the spring releases when it returns to its original state.
Work Done by Friction
Work done by friction occurs when a force opposes the movement of an object, transforming kinetic energy into heat or other forms of energy.
In the context of this problem, the block slides across a table and friction opposes its motion.
The force of friction is expressed as:
\[ f_k = \mu_k \cdot F_N \]where:
  • \( f_k \) is the force of friction
  • \( \mu_k \) is the coefficient of kinetic friction
  • \( F_N \) is the normal force, typically calculated as \( m \cdot g \) where \( m \) is mass and \( g \) is acceleration due to gravity
The work done by friction is the product of the frictional force and the distance over which it acts:
\[ \text{Work by friction} = f_k \cdot d \]In this scenario, since all potential energy in the spring transitions into work against friction, we equate the work done by friction to the spring's stored energy (2 J), allowing us to find \( \mu_k \).
Spring Constant
The spring constant \( k \) is a measure of a spring's stiffness, defined as the force required per unit displacement to compress or extend the spring.
A larger spring constant means a stiffer spring that requires more force to compress or stretch by a given amount.
In the discussed problem, the spring constant is given as \( 100 \, \text{N/m} \).
To see how this works in action, remember the spring potential energy formula, where \( k \) plays a critical role:
\[ U = \frac{1}{2} k x^2 \]Given the formula, changing \( k \) directly affects the amount of energy stored when the spring is compressed or extended.
A higher spring constant would increase the potential energy for the same displacement \( x \).
By understanding the spring constant, you can determine how different springs with varied lengths or materials might behave under similar conditions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

CALC An object has several forces acting on it. One force is \(\vec{\boldsymbol{F}}=\alpha x y \hat{\boldsymbol{r}},\) a force in the \(x\) -direction whose magnitude depends on the position of the object. (Sce Problem \(6.98 . )\) The constant is \(\alpha=2.00 \mathrm{N} / \mathrm{m}^{2}\) . The object moves along the following path: \((1)\) It starts at the origin and moves along the \(y\) -axis to the point \(x=0\) , \(y=1.50 \mathrm{m} ;(2)\) it moves parallel to the \(x\) -axis to the point \(x=1.50 \mathrm{m}, y=1.50 \mathrm{m} ;(3)\) it moves parallel to the \(y\) -axis to the point \(x=1.50 \mathrm{m}, y=0 ;(4)\) it moves parallel to the \(x\) -axis back to the origin. (a) Sketch this path in the \(x y\) -plane. (b) Calculate the work done on the object by \(\vec{F}\) for each leg of the path and for the complete round trip. (c) Is \(\vec{F}\) conservative or nonconservative? Explain.

Tarzan and Jane. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 \(\mathrm{m}\) that makes an angle of \(45^{\circ}\) with the vertical, steps off his tree limb, and swings down and then up to Jane's open arms. When he arrives, his vine makes an angle of \(30^{\circ}\) with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan's speed just before he reaches Jane. You can ignore air resistance and the mass of the vine.

The Great Sandini is a 60 -kg circus performer who is shot from a cannon (actually a spring gun). You don't find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100 \(\mathrm{N} / \mathrm{m} / \mathrm{m}\) that he will compress with a force of 4400 \(\mathrm{N}\) .The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40 \(\mathrm{N}\) during the 4.0 \(\mathrm{m}\) he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.5 \(\mathrm{m}\) above his initial rest position?

If a fish is attached to a vertical spring and slowly lowered to its equilibrium position, it is found to stretch the spring by an amount \(d .\) If the same fish is attached to the end of the unstretched spring and then allowed to fall from rest, through what maximum distance does it stretch the spring? (Hint: Calculate the force constant of the spring in terms of the distance \(d\) and the mass \(m\) of the fish.)

Cp A small block with mass 0.0500 kg slides in a vertical circle of radius \(R=0.800 \mathrm{m}\) on the inside of a circular track. There is no friction between the track and the block. At the bottom of the block's path, the normal force the track exerts on the block has magnitude 3.40 \(\mathrm{N} .\) What is the magnitude of the normal force that the track exerts on the block when it is at the top of its path?
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Fluids

Read Explanation

Work Energy and Power

Read Explanation

Electrostatics

Read Explanation

Measurements

Read Explanation

Atoms and Radioactivity

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.