1206 - Concepts in Physics

Friday, September 25

WORK and ENERGY• Today we will define work and energy

• In physics these terms mean very specific things

• They are not necessarily what our intuition tells us - so be careful

• We need to remember the difference between a vector and a scalar

What is work

• YOUR TURN:

• take a minute and write down a few things that come to mind, when you think work

• think along your life experiences ...

What is Work? The work done by an agent exerting a constant force F and causing a displacement s equals the magnitude of the displacement, times the component of F along the direction of s .

Note: • If s = 0 W = 0 no work is done when holding a heavy box, or pushing against a wall

• W = 0 if F ⊥ s no work is done by carrying a bucket of water horizontally

• The sign of W depends on the direction of F relative to s: W > 0 when component of F along s is in the same direction as s, and W < 0 when it is in the opposite direction. This sign is given automatically if we write as

the angle between F and s and write W = F s cosΘ .

• If F acts along the direction of s, then W = Fs , since cos Θ = cos 0 = 1.

• Work is a scalar.

• The SI units of work are Joules (J) (1 Joule = 1 Newton meter = 1 Nm).

W = F s cosΘ

Calculating the Amount of Work Done by Forces

When a force acts to cause an object to be displaced, three quantities must be known in order to calculate the work. Those three quantities are force, displacement and the angle between the force and the displacement. The work is subsequently calculated as force*displacement*cosine(theta) where theta is the angle between the force and the displacement vectors.

Apply the work equation to determine the amount of work done by the applied force in each of the three situations described below.

YOUR TURN:

Potential EnergyAn object can store energy as the result of its position. For example, the heavy heavy ball of a demolition machine is storing energy when it is held at an elevated position. This stored energy of position is referred to as potential energy. Similarly, a drawn bow is able to store energy as the result of its position. When assuming its usual position (i.e., when not drawn), there is no energy stored in the bow. Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position. Potential energy is the stored energy of position possessed by an object.

 

Gravitational potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object. The higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation:

PEgrav = mass * g * heightPEgrav = m * g * h

In the above equation, m represents the mass of the object, h represents the height of the object and g represents the acceleration of gravity (9.8 m/s/s on Earth).

Since the gravitational potential energy of an object is directly proportional to its height above the zero position, a doubling of the height will result in a doubling of the gravitational potential energy. A tripling of the height will result in a tripling of the gravitational potential energy.

YOUR TURN: Use this principle to determine the blanks in the following diagram. Knowing that the potential energy at the top of the tall platform is 50 J, what is the potential energy at the other positions shown on the stair steps and the incline?

Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy.

Springs are a special instance of a device which can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force which is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the constant of proportionality is known as the spring constant (k).

Fspring = k * x

Such springs are said to follow Hooke's Law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs which relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is

A cart is loaded with a brick and pulled at constant speed along an inclined plane to the height of a seat-top. If the mass of the loaded cart is 3.0 kg and the height of the seat top is 0.45 meters, then what is the potential energy of the loaded cart at the height of the seat-top?

YOUR TURN:

Kinetic Energy and the Work Energy TheoremIdea: Force is a vector, work and energy are scalars. Thus, it is often easier to solve problems using energy considerations instead of using Newton's laws (i.e. it is easier to work with scalars than vectors).

Definition:

The kinetic energy ( Ekin ) of an object of mass m that is moving with velocity v is:  

   

Note: • Kinetic energy is a scalar. • The units are the same as for work (i.e. Joules, J).

Relation bewteen Ekin and W: The work done on an object by a net force equals the change in kinetic energy of the object:

 

This relationship is called the Energy-Work theorem.

Ekin =½mv2

W = Ekin (final) - Ekin (initial)

Kinetic energy is the energy of motion. An object which has motion - whether it be vertical or horizontal motion - has kinetic energy. There are many forms of kinetic energy - vibrational (the energy due to vibrational motion), rotational (the energy due to rotational motion), and translational (the energy due to motion from one location to another).

The kinetic energy equation reveals that the mass of an object is directly proportional to the square of its speed. That means that for a twofold increase in speed, the kinetic energy will increase by a factor of four. For a threefold increase in speed, the kinetic energy will increase by a factor of nine. And for a fourfold increase in speed, the kinetic energy will increase by a factor of sixteen. The kinetic energy is dependent upon the square of the speed. As it is often said, an equation is not merely a recipe for algebraic problem-solving, but also a guide to thinking about the relationship between quantities.

Let’s take a moment and draw a linear and quadratic relationship ...

YOUR TURN:

1. Determine the kinetic energy of a 600-kg roller coaster car that is moving with a speed of 18 m/s.

2. If the roller coaster car in the above problem were moving with twice the speed, then what would be its new kinetic energy?

3. Missy Diwater, the former platform diver for the Ringling Brother's Circus, had a kinetic energy of 12 000 J just prior to hitting the bucket of water. If Missy's mass is 40 kg, then what is her speed?

In the process of doing work, the object which is doing the work exchanges energy with the object upon which the work is done. When the work is done upon the object, that object gains energy. The energy acquired by the objects upon which work is done is known as mechanical energy.

Mechanical energy

Mechanical energy is the energy which is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position).

A moving car possesses mechanical energy due to its motion (kinetic energy). A moving baseball possesses mechanical energy due to both its high speed (kinetic energy) and its vertical position above the ground (gravitational potential energy).

The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME).

TME = PE + KE

The diagram below depicts the motion of Li Ping Phar (esteemed Chinese ski jumper) as she glides down the hill and makes one of her record-setting jumps.

The total mechanical energy of Li Ping Phar is the sum of the potential and kinetic energies. The two forms of energy sum up to 50 000 Joules. Notice also that the total mechanical energy of Li Ping Phar is a constant value throughout her motion.

An object with mechanical energy is able to do work on another object.

YOUR TURN:

imaging the loop of a rooller coaster - at which point is the kinetic energy the

largest? Which point has the most potential

energy? What condition in terms of energy

budget needs to be fulfilled to ensure, that it doesn’t fall down?

Luisa and her friend Miguel have decided they want to hike up Pike’s Peak. Luisa has done some research and decided that they will go up the Barr Trail, then come part way back down and meet her dad at the Crags parking lot. The Barr Trail is 13 miles each direction with a vertical gain of about 7000 feet, and the trail back to the parking lot is about 6.5 miles and 4200 feet vertical. However, the path up contains numerous switchbacks that vary from very steep to completely flat. They also must work much harder to climb as they get up into the thinner air at higher altitudes. Luisa weighs about 140 lbs and Miguel weighs about 180 lbs, including equipment.

a. Approximately how much work against gravity did each of them do? b. Is this the same as the actual work they did? Why or why not? c. Where did the energy they used come from? d. What happened to the energy not used to do work against gravity?

1.) In Harry Potter and the Sorcerer’s Stone by J.K. Rowling, Ron Weasley manages to knock a troll out by levitating its club and then dropping the club on the troll’s head.

If the troll is 12 feet tall and Ron manages to drop the club, weighing 50 kg, from 15 feet in the air, how much kinetic energy will it transfer to the trolls head when it strikes?

2.) The first woman in space, cosmonaut Valentina Tereshkova, flew in 1963. She orbited the earth 48 times during her 70 hour 50 minute flight at an average altitude of 165.5 km above the surface of the earth. Her spacecraft, Vostok 6, had a mass of about 4700 kg.

a. What was her average kinetic energy in this flight? b. What was her average gravitational potential energy?