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 9

Chapter 10, Problem 9

A 20 kg child is on a swing that hangs from 3.0-m-long chains. What is her maximum speed if she swings out to a 45° angle?

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Determine the height difference between the highest point of the swing and the lowest point. Use trigonometry to calculate this. The height difference can be found using the formula: \( h = L - L \cos(\theta) \), where \( L \) is the length of the chain (3.0 m) and \( \theta \) is the angle (45 degrees).

Calculate the potential energy at the highest point of the swing relative to the lowest point. Use the formula: \( PE = mgh \), where \( m \) is the mass of the child (20 kg), \( g \) is the acceleration due to gravity (9.8 m/s^2), and \( h \) is the height difference calculated in the previous step.

At the lowest point of the swing, all the potential energy is converted into kinetic energy (assuming no energy loss due to air resistance or friction). Use the conservation of energy principle: \( PE_{\text{highest}} = KE_{\text{lowest}} \). The kinetic energy is given by \( KE = \frac{1}{2}mv^2 \), where \( v \) is the speed at the lowest point.

Rearrange the kinetic energy formula to solve for \( v \): \( v = \sqrt{2gh} \). Substitute the value of \( h \) from step 1 and \( g \) to find the maximum speed.

Perform the substitution and simplify the expression to find the numerical value of \( v \). This will give the maximum speed of the child at the lowest point of the swing.

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Key Concepts

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

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 the context of the swing, the potential energy at the highest point of the swing is converted into kinetic energy at the lowest point. This relationship allows us to calculate the maximum speed of the child by equating the potential energy at the 45-degree angle to the kinetic energy at the lowest point.

Potential Energy

Potential energy is the energy stored in an object due to its position or configuration. For the child on the swing, gravitational potential energy is given by the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above a reference point. When the swing is at a 45-degree angle, the height can be calculated using trigonometric functions, which is essential for determining the potential energy at that position.

Kinetic Energy

Kinetic energy is the energy of an object in motion, defined by the formula KE = 0.5mv², where m is mass and v is velocity. As the child swings down from the 45-degree angle to the lowest point, the potential energy is converted into kinetic energy, reaching its maximum at the lowest point. Understanding this concept is crucial for calculating the maximum speed of the child as she swings.

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