Problem 20 - Applied Physics

Question

A certain machine is designed to perform a given amount of work in a given amount of time. A second machine does \(2.5\) times the same amount of work in one third the time. Find the power of the second machine compared with the first.

Step-by-Step Solution

Answer

The second machine's power is 7.5 times that of the first machine.

Step 1: Understanding Power Formula

Power is defined as the work done per unit of time. The formula for power is given by:\[ P = \frac{W}{t} \]where \( P \) is power, \( W \) is work done, and \( t \) is the time taken.

Step 2: Analyzing the First Machine

Let's assume the first machine does \( W \) work in \( t \) time. Therefore, the power of the first machine is:\[ P_1 = \frac{W}{t} \]

Step 3: Analyzing the Second Machine

The second machine does \( 2.5W \) in \( \frac{t}{3} \) time. Thus, the power of the second machine is:\[ P_2 = \frac{2.5W}{\frac{t}{3}} = \frac{2.5W \times 3}{t} = \frac{7.5W}{t} \]

Step 4: Calculating Power Ratio

To find how the power of the second machine compares with the first, we calculate the ratio of their powers:\[ \text{Power ratio} = \frac{P_2}{P_1} = \frac{\frac{7.5W}{t}}{\frac{W}{t}} = 7.5 \]

Key Concepts

Work and Energy

Work and energy are fundamental concepts in physics, often used to describe the ability to apply force and move objects, regardless of whether it's mechanical or otherwise. **Work** is defined when a force is applied to an object and this object moves in the direction of the applied force. If there's no movement, no work is done, even if a force is applied.
The formula for work is:

• \( W = F \cdot d \)

Where \( W \) represents the work done, \( F \) is the force applied, and \( d \) is the displacement in the direction of the force.**Energy**, on the other hand, is the capacity to do work. It exists in different forms, such as kinetic energy, potential energy, thermal energy, etc. When you do work on an object, such as lifting a book on a shelf, you are transferring energy to it. This energy becomes the stored potential energy when the book is at a height. If the book falls, this energy changes into kinetic energy as it moves downwards. Understanding the connection between work and energy helps in solving problems related to motion and force.

Time and Speed Relationships

Understanding the relationship between time and speed is essential when analyzing how machines work, as seen in our exercise. **Speed** is how fast something is moving, while **time** is how long it's taking to do so. In a physics context, if a machine or an object travels a certain distance, you can determine its speed with the formula:

• \( ext{Speed} = \frac{ ext{Distance}}{ ext{Time}} \)

If a machine needs to accomplish more work in less time, like the second machine in our problem, it means it must operate at a faster speed. In our exercise, although both machines do the same amount of work, the second machine completes its work quicker. This adjustment in time and speed influences power--the energy converted per unit time. Faster speeds mean larger amounts of energy are being moved or processed in less time, which explains why the second machine has greater power.

Mechanical Advantage

Mechanical advantage gives insight into the efficiency and usefulness of machines. It describes how a machine can amplify an input force to produce a greater output force. For simple machines, this factor determines how work input can be minimized by increasing the distance the input force travels or by increasing the magnitude of the input force.Mathematically, mechanical advantage (MA) is expressed as:

• \( ext{MA} = \frac{ ext{Output Force}}{ ext{Input Force}} \)

In the context of the given problem, though mechanical advantage isn’t directly involved, the concept tells us that the second machine, by being more powerful, carries out the work more efficiently. When thinking about two machines doing the same task but with different power outputs, the one with greater power might be seen as having a practical mechanical advantage. This is because it can accomplish the same work faster with more power exerted over time, implying a greater force or speed which is advantageous in many operational contexts.