Textbook Question
A -kg box moving at m/s on a horizontal, frictionless surface runs into a light spring of force constant N/cm. Use the work–energy theorem to find the maximum compression of the spring.
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Ch 06: Work & Kinetic Energy
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
Chapter 6, Problem 54
A -kg rock is sliding on a rough, horizontal surface at m/s and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is . What average power is produced by friction as the rock stops?
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Step 1: Identify the forces acting on the rock. The force of kinetic friction is given by \( F_{\text{friction}} = \mu_k \cdot F_{\text{normal}} \), where \( \mu_k \) is the coefficient of kinetic friction and \( F_{\text{normal}} \) is the normal force. Since the surface is horizontal, \( F_{\text{normal}} = m \cdot g \), where \( m \) is the mass of the rock and \( g \) is the acceleration due to gravity.
Step 2: Calculate the work done by friction. Work is defined as \( W = F_{\text{friction}} \cdot d \), where \( d \) is the distance over which the force acts. To find \( d \), use the kinematic equation \( v^2 = u^2 + 2a \cdot d \), where \( v \) is the final velocity (0 m/s), \( u \) is the initial velocity (8.00 m/s), and \( a \) is the acceleration caused by friction. The acceleration can be found using \( a = \frac{F_{\text{friction}}}{m} \).
Step 3: Determine the time it takes for the rock to stop. Use the kinematic equation \( v = u + a \cdot t \), where \( t \) is the time, \( v \) is the final velocity (0 m/s), \( u \) is the initial velocity (8.00 m/s), and \( a \) is the acceleration caused by friction.
Step 4: Calculate the average power produced by friction. Power is defined as \( P = \frac{W}{t} \), where \( W \) is the work done by friction and \( t \) is the time it takes for the rock to stop. Substitute the values of \( W \) and \( t \) obtained from the previous steps into this formula.
Step 5: Verify the units and ensure consistency. Work is measured in joules (\( J \)), time in seconds (\( s \)), and power in watts (\( W \)). Confirm that all calculations align with these units and the physical principles involved.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Friction
Friction is a force that opposes the relative motion of two surfaces in contact. In this scenario, kinetic friction acts on the rock as it slides, causing it to decelerate and eventually stop. The magnitude of kinetic friction can be calculated using the formula F_friction = μ_k * N, where μ_k is the coefficient of kinetic friction and N is the normal force.
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Work-Energy Principle
The work-energy principle states that the work done by all forces acting on an object equals the change in its kinetic energy. As the rock slides to a stop, the work done by friction is equal to the initial kinetic energy of the rock, which can be calculated using KE = 0.5 * m * v^2, where m is mass and v is velocity.
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Power
Power is defined as the rate at which work is done or energy is transferred over time. It can be calculated using the formula P = W/t, where W is work and t is time. In this context, the average power produced by friction can be determined by dividing the work done by friction (which equals the initial kinetic energy of the rock) by the time it takes for the rock to come to a stop.
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Related Practice
Textbook Question
A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a -cm strip of the donated aorta reveal that it stretches cm when a -N pull is exerted on it. What is the force constant of this strip of aortal material?
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Textbook Question
A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a -cm strip of the donated aorta reveal that it stretches cm when a -N pull is exerted on it. If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is cm, what is the greatest force it will be able to exert there?
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