91Ó°ÊÓ

When we think about doing work in physics, it means that a force is acting over a distance. Power, on the other hand, is all about how quickly this work is done. It measures the rate of doing work or transferring energy.
For example, if you push a shopping cart, the work done is the force you apply multiplied by the distance the cart travels. Power would then measure how fast you moved the cart. If you push it quickly, you're using more power.
In mathematical terms, work is given by the equation \( ext{Work} = ext{Force} \times ext{Distance} \), and power is defined as \( ext{Power} = \frac{ ext{Work}}{ ext{Time}} \). In our exercise, the work done by the motor is the energy used to heat the water. The motor's power is 0.4 hp, which is a measure of how much work it can do in a certain time. To find out how long this work takes, we use the formula \( ext{Power} \times ext{Time} = ext{Work} \).
This relationship helps us understand that increasing the motor's power, or the time it runs, will increase the work done and consequently more energy transferred into heating the water.

Converting units is like translating languages; you're expressing the same quantity in different terms. This is crucial in physics because it helps us use formulas correctly and compare values accurately.
In the exercise, we convert power from horsepower to watts because the SI unit of power is the watt, not horsepower. We also convert thermal energy from calories to joules since energy calculations often use joules. One calorie is equal to 4.184 joules.

• To convert horsepower to watts, we multiply by 746 (since 1 hp = 746 Watts).
• To convert kcal to joules, we multiply by 4.184.

Unit conversions help ensure all parts of a calculation are consistent with each other. This consistency is important for accurate calculation and understanding of the problem. Whenever you're stuck with different units, remember it's about finding a common language for the problem.

Motor efficiency refers to how well a motor converts electrical energy into mechanical energy. In an ideal world, all the energy used by a motor would go into doing the work we want. But reality includes some losses, like heat energy due to friction, which is evident in our exercise where energy from the motor is used to heat the water.
Efficiency is mathematically expressed as \( ext{Efficiency} = \frac{ ext{Useful Work Output}}{ ext{Total Energy Input}} \times 100\% \). In this case, all the work done by the motor is assumed to go into heating the water – an ideal scenario where the efficiency for this task is 100%.
However, typically, motors are not perfectly efficient due to energy lost as heat and to overcoming friction. The purpose of calculating efficiency is to understand how much energy is effectively used for the designed purpose versus how much is wasted. This helps in designing better motors and improving energy conservation in appliances and industrial applications.