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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
All textbooksKnight Calc 5th EditionCh 10: Interactions and Potential EnergyProblem 41
Chapter 10, Problem 41

A 50 g mass is attached to a light, rigid, 75-cm-long rod. The other end of the rod is pivoted so that the mass can rotate in a vertical circle. What speed does the mass need at the bottom of the circle to barely make it over the top of the circle?

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Step 1: Identify the forces acting on the mass at the top of the circle. At the top, the gravitational force and the centripetal force must be balanced for the mass to barely make it over the top. The centripetal force is provided entirely by gravity in this case.
Step 2: Write the condition for the mass to barely make it over the top of the circle. The centripetal force required at the top is given by \( F_c = \frac{m v_{top}^2}{r} \), where \( m \) is the mass, \( v_{top} \) is the speed at the top, and \( r \) is the radius of the circle. For the mass to barely make it over, \( v_{top} \) must be such that \( F_c = mg \), where \( g \) is the acceleration due to gravity.
Step 3: Solve for \( v_{top} \). Using \( F_c = mg \), substitute \( mg = \frac{m v_{top}^2}{r} \). Cancel \( m \) from both sides to get \( v_{top}^2 = gr \). Take the square root to find \( v_{top} = \sqrt{gr} \).
Step 4: Use conservation of mechanical energy to relate the speed at the bottom of the circle \( v_{bottom} \) to the speed at the top \( v_{top} \). At the bottom, the mass has both kinetic energy and potential energy, while at the top, it has only potential energy. Write the energy conservation equation: \( KE_{bottom} + PE_{bottom} = KE_{top} + PE_{top} \). Substitute \( KE = \frac{1}{2}mv^2 \) and \( PE = mgh \), where \( h \) is the height relative to the bottom.
Step 5: Solve for \( v_{bottom} \). At the bottom, \( h = 0 \), so \( PE_{bottom} = 0 \). At the top, \( h = 2r \), so \( PE_{top} = mg(2r) \). Substitute \( KE_{bottom} = \frac{1}{2}m v_{bottom}^2 \), \( KE_{top} = \frac{1}{2}m v_{top}^2 \), and \( PE_{top} = mg(2r) \) into the energy conservation equation: \( \frac{1}{2}m v_{bottom}^2 = \frac{1}{2}m v_{top}^2 + mg(2r) \). Cancel \( m \) from all terms and solve for \( v_{bottom} \).

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Key Concepts

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

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For an object in vertical circular motion, this force is provided by the gravitational force and the tension in the rod. At the top of the circle, the gravitational force must be sufficient to provide the necessary centripetal force to keep the mass in circular motion.
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Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. In the context of circular motion, as the mass rises to the top of the circle, its GPE increases while its kinetic energy decreases. To just make it over the top, the mass must have enough speed at the bottom to convert kinetic energy into GPE at the top.
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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 scenario, the total mechanical energy (kinetic plus potential) of the mass must remain constant throughout its motion. This means that the kinetic energy at the bottom of the circle must equal the sum of the potential energy at the top and any remaining kinetic energy needed to maintain circular motion.
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Related Practice
Textbook Question

You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 10-m-high hill, then descends 15 m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 400 kg. For safety reasons, the spring constant should be 10% larger than the minimum needed for the car to just make it over the top. What is the maximum speed of a 350 kg car if the spring is compressed the full amount?

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Textbook Question

A particle moves from A to D in FIGURE EX10.36 while experiencing force F = (6i + 8j) N. How much work does the force do if the particle follows path ACD. Is this a conservative force? Explain.

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Textbook Question

A block of mass m slides down a frictionless track, then around the inside of a circular loop-the-loop of radius R . From what minimum height h must the block start to make it around without falling off? Give your answer as a multiple of R.

Textbook Question

How much work is done by the environment in the process shown in FIGURE EX10.39? Is energy transferred from the environment to the system or from the system to the environment?

Textbook Question

A 50 g ice cube can slide up and down a frictionless 30° slope. At the bottom, a spring with spring constant 25 N/m is compressed 10 cm and used to launch the ice cube up the slope. How high does it go above its starting point?

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Textbook Question

A cable with 20.0 N of tension pulls straight up on a 1.50 kg block that is initially at rest. What is the block's speed after being lifted 2.00 m? Solve this problem using work and energy.