# Physics. Energy and Power

###### tags: `Physics`

## Simple definition

(this is the formula that you get in the exam)

$$P = W/t$$

What does it mean?

Power means _rate of working_ in the same sense that velocity is _rate of movement_ and acceleration is _rate of velocity_.

We can see it as Work over time, in other words $P (power)= {W}/ {t}$

### Other units of Power

For the exam:

You need to know that the unit of power is Watt (W) and work with their compounds

$1GW = 1\ 000\ 000\ 000\ W = 10^9 W$
$1MW = 1\ 000\ 000\ W = 10^6 W$
$1KW = 1\ 000\ W = 10^3 W$ (beware of this one, don't mess with $KWh$)
$1mW = 0.\ 001 W = 10^{-3} W$

Also remember that we can use the term KWh that it's to measure energy, not power.

Not for the exam but for your information:

There is another unit, imperial HorsePower (HP) and metric HorsePower (CV) that are used, specially in cars.

1 metric hp = 735.5 W
1 imperial hp = 747.7 W



## The other way to get to power

The expression for power is work/time. And since the expression for work is force*displacement, the expression for power can be rewritten as (force*displacement)/time. Since the expression for velocity is displacement/time, the expression for power can be rewritten once more as force*velocity. This is shown below.


This new equation for power reveals that a powerful machine is both strong (big force) and fast (big velocity). A powerful car engine is strong and fast. A powerful piece of farm equipment is strong and fast. A powerful weightlifter is strong and fast. A powerful lineman on a football team is strong and fast. A machine that is strong enough to apply a big force to cause a displacement in a small mount of time (i.e., a big velocity) is a powerful machine.


([source](https://www.physicsclassroom.com/class/energy/Lesson-1/Power))

## Exercises!

### Easy solved exercise

(chin-up)

When doing a chin-up, a student lifts her 62.0 kg body a distance of 0.25 meters in 2 seconds. What is the power delivered by the student's biceps?

$W = F \Delta x \cos \alpha$
($\alpha$ is 0 because the force to put it up and the lift are in the same direction, so $\cos \alpha = 1$)

W = 62 kg * 9.81 m/s2 * 0.25 m

W = 152.1 J

P = W/t

P = 152.1 J / 2 s

P = 76.03 W

([source](https://courses.fortlewis.edu/courses/17334/pages/work-energy-and-power))

### 1 Derived exercise

Another studing it's also doing a chin up. In this case his body has a mass of 70 kg, the distance is 0.28 meters and the lift takes place in 2.5 seconds.

What is the power by the second student's biceps?

### 2 Exercise in the other way around

We have a robot that it's going to do the same chin-up.

Data:

The robot has 3 bolts
The lift is 0.30 meters
The pneumatic motors of the robots arms have a combined a power of 150 W
The robot has a mass of 100 kg

How long is it going to take to the robot to make the chin-up?

What is the name of the robot?

#### Hint for exercise 2.

First we need to know how much work is going to do in Joules, to find it we have to use the mass and the distance.

After that we have 2 of the 3 elements of the formula P = W /t, we need to find the third one (time).

(for the name, be creative)

### Exercise with ramp and human (solved)


A 70 kg cyclist is riding their 5 kg bike up a 10º slope at 5 m/s. What is the mechanical power of the cyclist?

First we find out the F down

Fdown = sin10o * (70 kg + 5 kg)* 9.81 m/s2

Fdown = 127.8 N

Then we apply P = Fv

Pmech = Fv

Pmech = 127.8 N * 5 m/s

Pmech = 639 W

### 3 Derived exercise

The same cyclist is riding a 4 kg bike but the slope is more tight, it's a a 12º slope at 5 m/s. What is the mechanical power of this cyclist? (consider the slope of 12º)

### 4 Exercise in the other way around

We built another robot (this time a bicycle robot) with a power of 1kW and a total mass of 80 kg. What would be its speed? What is its name?

# Efficiency

We make machines and we can be considered also a machine that has some input and some outputs. From the input to the output we will lose some energy in the process (friction, joule effect, heat, tension). So, for example if the robot that you named in exercise #2 had a power of 150 W, it's consuption from the electric grid will be higher. To measure it we use **efficiency**. It's a ratio.

In short:

$$Efficiency = \frac{output\ Power}{input\ Power}$

Efficiency will be always between 0 and 1 or, if we use percentages, between 0 and 100.

The questions are usually about

I have

## Exercises!

Usually efficiency is not addressed in their own but as a part of a bigger exercise with power and work.

### Solved exercise

For example let's go back to the cyclist.

We have the concept of _metabolic power_ that is the power that uses your human body to do a mechanical task. (some movement task, I mean, doesn't have to be a boring/repetitive one). It's the power that your cells consume for doing something.

We have an estimate that the human efficency is 26 %. **What is the metabolic power done by the cyclist? **

Efficiency = Pmech/Pmet

26% = Pmech/Pmet

Pmet = 639 W /0.26 = 2,458 W

### 5 Derived exercise

Assuming the same efficency, what would be the metabolic power of the cyclist in the exercise 3?

### 6 Exercise in the other way around

The cyclist robot has a power consumption of 80 KJ in 1 minute. What is the efficiency of its motors?

#### Hint for exercise 6.

See that here we don't have directly the power consumption but the energy consumption. We have to find power knowing that P=W/t. Remember to change units to have J and seconds.

Then we use the found power and we make the ratio (remember to use in both the same units!)

### Exam like questions:

An escalator is used to move 20 passengers every minute from the first floor of a department store to the second. The second floor is located 5.20 meters above the first floor. The average passenger's mass is 54.9 kg. Determine the power requirement of the escalator in order to move this number of passengers in this amount of time.

([source](https://www.physicsclassroom.com/class/energy/Lesson-1/Power))

---

During a physics lab, Jack and Jill ran up a hill. Jack is twice as massive as Jill; yet Jill ascends the same distance in half the time. Who did the most work? ______________ Who delivered the most power? ______________ Explain your answers.

([source](https://www.physicsclassroom.com/class/energy/Lesson-1/Power))

---

4. Hiking Colorado’s 14,000 foot (4000m) mountain peaks is a popular activity. A trail up Long’s Peak climbs from 9,000 ft to 14,000 ft in 6.2 miles, or in metric units, 1500m vertical in 10,000m.
a. How much mechanical work against gravity would a 70kg person perform climbing Long’s Peak?

b. In summer it is important to not be atop a mountain during the afternoon when lightning storms abound. A common “rule of thumb” for estimating how long a mountain hike will take is that “every 1000 feet (300m) of vertical takes an hour”. Given this estimate, what would the person’s average power output against gravity be during the climb up Long’s Peak?

([source](https://courses.fortlewis.edu/courses/17334/assignments/277693?module_item_id=491774))

---

18. Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, it is a zero-watt device.) Explain in terms of the definition of power.

([source](https://phys.libretexts.org/Bookshelves/University_Physics/Exercises_(University_Physics)/Exercises%3A_College_Physics_(OpenStax)/07%3A_Work_Energy_and_Energy_Resources_(Exercises)))