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Chapter 6: Problem 62

A person is making homemade ice cream. She exerts a force of magnitude \(22 \mathrm{N}\) on the free end of the crank handle on the ice-cream maker, and this end moves on a circular path of radius \(0.28 \mathrm{m}\). The force is always applied parallel to the motion of the handle. If the handle is turned once every \(1.3 \mathrm{s},\) what is the average power being expended?

Short Answer

Expert verified
The average power expended is approximately 29.8 W.

Step by step solution

01

Calculate the Work Done per Turn

To find the work done in one complete turn, we use the formula for work done in circular motion: \( W = F \times d \), where \( F \) is the force and \( d \) is the distance traveled in one complete turn (the circumference of the circle). The distance \( d = 2 \pi r \), where \( r = 0.28 \) m. \[ d = 2 \pi \times 0.28 \approx 1.76 \text{ m} \]Now, the work done per turn:\[ W = 22 \times 1.76 = 38.72 \text{ J} \]
02

Calculate the Time for One Turn

The problem states that the handle is turned once every \(1.3\) seconds. This means the time taken for one complete turn is \( t = 1.3 \text{ s} \).
03

Calculate the Average Power Expended

Power is the rate at which work is done, given by \( P = \frac{W}{t} \), where \( W \) is the work per turn and \( t \) is the time for one turn. Using the previously found values:\[ P = \frac{38.72}{1.3} \approx 29.78 \text{ W} \]
04

Round Off the Answer

Round the calculated power to a suitable number of significant figures based on the inputs:\( P \approx 29.8 \text{ W} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circular Motion
Circular motion occurs when an object follows a path along a circular trajectory. Imagine the path that a person takes when turning the handle of the ice-cream maker鈥攊t鈥檚 a perfect circle! This is a classic example of circular motion. Here, the crank handle moves in such a way that its endpoint follows the circumference of the circle. Key points to understand about circular motion:
  • Path: The movement follows a circular path.
  • Radius: The radius remains constant, meaning the distance from the circle's center to the edge is always the same (in this exercise, it鈥檚 0.28 meters).
  • Uniform motion: In many problems, including this one, it鈥檚 assumed that the motion is uniform, which means the speed of rotation doesn鈥檛 change.
This simple scenario gives us a foundation for calculating work done, which is a vital part of physics. When dealing with circular motion, understanding the path and how forces act upon objects will help visualize and solve problems more easily.
Work Done
Work done is a concept in physics that measures the amount of energy transferred by a force acting over a distance. In simpler terms, it's how much 'push' you give an object times how far you push it. In the context of the ice-cream maker exercise, work done (\(W\)) can be calculated using the formula \[W = F \times d\]where \(F\) is the force, and \(d\) is the distance the handle travels. Here are some useful tips:
  • Force: It should be in the direction of motion (22 N in this exercise).
  • Distance: In a circle, this would be the circumference which is \(2 \pi r\).
The formula shows us that work is only done when the object moves along the direction of the force. In our exercise, as the force is applied parallel to the motion, work is done every time the handle completes a circle.
Force in Physics
Force is a push or pull exerted on an object, often causing it to move. In the physical world, force is essential as it helps us explain how objects interact. In the ice-cream making scenario, the force applied is the effort by which the person turns the crank handle. Understanding the idea of force involves these aspects:
  • Magnitude: This tells us the strength of the force. In the exercise, it is given as 22 N (Newtons).
  • Direction: Force has a direction, and in our problem, it鈥檚 crucial because it aligns with the motion of the handle.
  • Resulting motion: When the applied force remains consistent, the handle continues its circular motion effectively and efficiently.
Forces allow us to understand changes in motion by helping us calculate work done and the consequent power in various situations.
Distance Traveled
Distance traveled refers to the total path length covered by an object during motion. In circular motion, like with the ice-cream maker's crank handle, the distance traveled for one complete turn of the circle is called the circumference.Here鈥檚 what you should know:
  • The formula for the circumference of a circle is \(d = 2 \pi r\), where \(r\) is the radius of the circle.
  • In our case, since the radius is 0.28 m, the distance the handle covers in one turn is approximately 1.76 m.
  • This distance is crucial for calculating the work done since it's the path length along which the force is applied.
Understanding distance in physics helps you determine how far an object moves under the influence of a force, which in turn connects to the calculation of work done and power. With each turn, knowing how far the handle moves validates all calculations related to effort and energy expenditure.

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Most popular questions from this chapter

A \(1.00 \times 10^{2}-\mathrm{kg}\) crate is being pushed across a horizontal floor by a force \(\overrightarrow{\mathbf{P}}\) that makes an angle of \(30.0^{\circ}\) below the horizontal. The coefficient of kinetic friction is \(0.200 .\) What should be the magnitude of \(\overrightarrow{\mathbf{P}},\) so that the net work done by it and the kinetic frictional force is zero?

"Rocket Man" has a propulsion unit strapped to his back. He starts from rest on the ground, fires the unit, and accelerates straight upward. At a height of \(16 \mathrm{m},\) his speed is \(5.0 \mathrm{m} / \mathrm{s} .\) His mass, including the propulsion unit, has the approximately constant value of 136 kg. Find the work done by the force generated by the propulsion unit.

? 67.0-kg person jumps from rest off a 3.00-m-high tower straight down into the water. Neglect air resistance. She comes to rest \(1.10 \mathrm{m}\) under the surface of the water. Determine the magnitude of the average force that the water exerts on the diver. This force is nonconservative.

A person pushes a 16.0-kg shopping cart at a constant velocity for a distance of \(22.0 \mathrm{m}\). She pushes in a direction \(29.0^{\circ}\) below the horizontal. A \(48.0-\mathrm{N}\) frictional force opposes the motion of the cart. (a) What is the magnitude of the force that the shopper exerts? Determine the work done by (b) the pushing force, (c) the frictional force, and (d) the gravitational force.

A 75.0 -kg man is riding an escalator in a shopping mall. The escalator moves the man at a constant velocity from ground level to the floor above, a vertical height of \(4.60 \mathrm{m}\). What is the work done on the man by (a) the gravitational force and (b) the escalator?
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