Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

Problem 52a

Chapter 10, Problem 52a

A freight company uses a compressed spring to shoot 2.0 kg packages up a 1.0-m-high frictionless ramp into a truck, as FIGURE P10.52 shows. The spring constant is 500 N/m and the spring is compressed 30 cm. What is the speed of the package when it reaches the truck?

Verified step by step guidance

Step 1: Identify the energy transformations in the system. The spring's potential energy is converted into the package's kinetic energy and gravitational potential energy as it moves up the ramp. Use the principle of conservation of energy to solve the problem.

Step 2: Write the formula for the spring's potential energy: \( U_s = \frac{1}{2} k x^2 \), where \( k \) is the spring constant (500 N/m) and \( x \) is the compression distance (0.30 m). Substitute these values to calculate the spring's potential energy.

Step 3: Write the formula for gravitational potential energy: \( U_g = m g h \), where \( m \) is the mass of the package (2.0 kg), \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height of the ramp (1.0 m). Substitute these values to calculate the gravitational potential energy.

Step 4: Apply the conservation of energy principle: \( U_s = K + U_g \), where \( K \) is the kinetic energy of the package at the top of the ramp. Rearrange the equation to solve for \( K \): \( K = U_s - U_g \). Use the values calculated in Steps 2 and 3.

Step 5: Use the formula for kinetic energy: \( K = \frac{1}{2} m v^2 \), where \( v \) is the speed of the package. Rearrange the equation to solve for \( v \): \( v = \sqrt{\frac{2K}{m}} \). Substitute the value of \( K \) from Step 4 and the mass of the package to find the speed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Potential Energy of a Spring

The potential energy stored in a compressed spring is given by the formula PE_spring = (1/2)kx^2, where k is the spring constant and x is the compression distance. In this scenario, the spring constant is 500 N/m and the spring is compressed by 0.3 m, allowing us to calculate the energy available to propel the package.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this problem, the potential energy stored in the spring is converted into kinetic energy as the package moves up the ramp, allowing us to relate the energies to find the speed of the package.

Kinetic Energy

Kinetic energy (KE) is the energy of an object due to its motion, calculated using the formula KE = (1/2)mv^2, where m is the mass and v is the velocity. To find the speed of the package when it reaches the truck, we will equate the kinetic energy to the potential energy converted from the spring and the gravitational potential energy gained while moving up the ramp.

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Textbook Question

A horizontal spring with spring constant 100 N/m is compressed 20 cm and used to launch a 2.5 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Use work and energy to find how far the box slides across the rough surface before stopping.

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Textbook Question

The spring shown in FIGURE P10.54 is compressed 50 cm and used to launch a 100 kg physics student. The track is frictionless until it starts up the incline. The student's coefficient of kinetic friction on the 30° incline is 0.15. What is the student's speed just after losing contact with the spring?

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Textbook Question

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Textbook Question

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