Domanda del libro di testo
A -kg book is sliding along a rough horizontal surface. At point it is moving at m/s, and at point it has slowed to m/s. How much work was done on the book between and ?
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Ch 06: Work & Kinetic Energy
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552
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TOC.tableOfContents
- Ch 01: Units, Physical Quantities & Vectors37
- Ch 02: Motion Along a Straight Line57
- Ch 03: Motion in Two or Three Dimensions45
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws54
- Ch 06: Work & Kinetic Energy11
- Ch 07: Potential Energy & Conservation6
- Ch 08: Momentum, Impulse, and Collisions15
- Ch 09: Rotation of Rigid Bodies37
- Ch 10: Dynamics of Rotational Motion44
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics27
- Ch 13: Gravitation34
- Ch 14: Periodic Motion26
- Ch 15: Mechanical Waves22
- Ch 16: Sound & Hearing28
- Ch 17: Temperature and Heat28
- Ch 18: Thermal Properties of Matter35
- Ch 19: The First Law of Thermodynamics29
- Ch 20: The Second Law of Thermodynamics18
- Ch 21: Electric Charge and Electric Field28
- Ch 22: Gauss' Law21
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF24
- Ch 26: Direct-Current Circuits29
- Ch 27: Magnetic Field and Magnetic Forces16
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction22
- Ch 30: Inductance14
- Ch 31: Alternating Current20
- Ch 33: The Nature and Propagation of Light17
- Ch 34: Geometric Optics29
- Ch 35: Interference13
- Ch 36: Diffraction19
- Ch 37: Special Relativity19
- Ch 38: Photons: Light Waves Behaving as Particles12
- Ch 39: Particles Behaving as Waves25
- Ch 40: Quantum Mechanics I: Wave Functions20
- Ch 41: Quantum Mechanics II: Atomic Structure21
- Ch 42: Molecules and Condensed Matter17
- Ch 43: Nuclear Physics13
- Ch 44: Particle Physics and Cosmology17
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Capitolo 6, Problema 37
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.
Guida verificata passo dopo passo1
Convert the spring constant from N/cm to N/m for consistency in SI units. Since 1 cm = 0.01 m, multiply the given spring constant (75 N/cm) by 100 to get the spring constant in N/m: \( k = 75 \times 100 = 7500 \ \text{N/m} \).
Use the work-energy theorem, which states that the work done on the box is equal to its change in kinetic energy. The box's initial kinetic energy is given by \( KE = \frac{1}{2} m v^2 \), where \( m = 6.0 \ \text{kg} \) and \( v = 3.0 \ \text{m/s} \). Substitute these values into the formula to calculate the initial kinetic energy.
At maximum compression of the spring, all the initial kinetic energy of the box is converted into elastic potential energy stored in the spring. The elastic potential energy of the spring is given by \( PE = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the maximum compression of the spring.
Set the initial kinetic energy of the box equal to the elastic potential energy of the spring: \( \frac{1}{2} m v^2 = \frac{1}{2} k x^2 \). Cancel out the \( \frac{1}{2} \) on both sides of the equation to simplify: \( m v^2 = k x^2 \).
Solve for \( x \), the maximum compression of the spring, by rearranging the equation: \( x = \sqrt{\frac{m v^2}{k}} \). Substitute the values \( m = 6.0 \ \text{kg} \), \( v = 3.0 \ \text{m/s} \), and \( k = 7500 \ \text{N/m} \) into the formula to find \( x \).
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Work-Energy Theorem
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this scenario, the kinetic energy of the box will be converted into potential energy stored in the spring as it compresses. This principle allows us to relate the initial kinetic energy of the box to the maximum compression of the spring.
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Percorso guidato
04:10
The Work-Energy Theorem
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v^2, where m is mass and v is velocity. For the box in the question, its initial kinetic energy can be determined using its mass (6.0 kg) and velocity (3.0 m/s), which will be crucial for calculating how much energy is transferred to the spring.
Video consigliato:
Percorso guidato
06:07Intro to Rotational Kinetic Energy
Spring Potential Energy
Spring potential energy is the energy stored in a compressed or stretched spring, given by the formula PE = 0.5 * k * x^2, where k is the spring constant and x is the compression or extension from its equilibrium position. In this case, the spring constant is provided (75 N/cm), and we will use this to find the maximum compression of the spring when the kinetic energy of the box is fully converted into spring potential energy.
Video consigliato:
Percorso guidato
06:18Energy in Horizontal Springs
Pratica correlata
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You throw a -N rock vertically into the air from ground level. You observe that when it is m above the ground, it is traveling at m/s upward. Use the work–energy theorem to find its maximum height.
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Domanda del libro di testo
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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Domanda del libro di testo
You throw a -N rock vertically into the air from ground level. You observe that when it is m above the ground, it is traveling at m/s upward. Use the work–energy theorem to find the rock's speed just as it left the ground.
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