Heat Transfer – A Nice Cup of Tea

Heat – or thermal energy –  is always transferred from hot to cold. More precisely, energy is transferred faster from hot to cold than it is from cold to hot.

If a hot cup of tea is left for a while, the tea goes cold, or the tea, the cup and the surroundings are all in thermal equilibrium. There are FOUR ways in which this happens:

1. The thermal energy from the tea is conducted through the walls of the cup.

2. The hot outer surface of the cup radiates heat into the air, heating it up.

3. Convection currents are set up in the surrounding air around the cup, the hot air rising above the cup and being replaced by colder air from below.

4. The molecules on the surface of the tea escape from the surface by evaporation – blowing over the surface makes this happen faster and the tea will cool more quickly.

To think about: design a teacup which will keep the tea hottest for the longest time.

Resistance and Ohm’s Law

The resistance of an electrical component tells us how many volts I need to push a current of 1A through it. It is measured in OHMS

1 Ohm = 1Volt per Amp

If we want to measure the resistance of a component, we connect up the circuit as shown.

The voltmeter is connected across the component and the ammeter can be anywhere in the loop. We take a set of values of V and plot them against I. If the V/I graph is a straight line, this means that resistance of the component doesn’t change and the component obeys Ohm’s Law. We can use any pair of values to find the resistance. You shouldn’t think of it as the slope of the line or in this case, the inverse of the slope, just select a convenient point on the line instead, read off V and I and divide one by the other.

Here’s some values to show you…The resistance = 12V / 3A = 4  ohms

I’ve put in a variable resistor so I can change the current in the circuit.

Ohm's Law is obeyed because the graph is a straight line

Electromotive force

….isn’t a force at all. It’s better described as ‘electrical pressure’ which we measure , like p.d, in VOLTS.

Defined as “power per unit current of a device or source of EMF” Be careful – EMF is thus the energy supplied to electric charge, POTENTIAL DIFFERENCE is energy dissipated (given up) by charge passing through resistance. 

It’s written on the battery for you. These are both 1.5V cells, connected in series. Notice we connect + to –

Two 1.5V cells connected in series. EMF= 3V

When the cells aren’t connected to anything – in other words they’re not pushing electrons round a circuit – a voltmeter connected across the ends will give you a value for EMF.

As soon as the cells are connected into a circuit and energy is drawn from them, the voltage measured across the cells will fall because we need a little bit of energy per coulomb (a few volts) to push charge through the battery itself. We say that the battery has an INTERNAL RESISTANCE. We can’t get back or recover these LOST VOLTS across the battery. When a car battery is getting old, its internal resistance increases so it needs a lot of energy to push charge through it, the lost volts increases so it can’t deliver the required 12V to an external circuit and needs to be replaced.

Because voltage is a measure of energy per coulomb, and energy can’t just be ‘lost’, the voltages across each component add up around a circuit to equal the supply voltage.

Projectiles

The problem of the ‘Monkey and the Hunter’ is a famous one in physics. Here’s the picture.

The Monkey and the Hunter

Where should the hunter aim to be sure of hitting the monkey, at him, above him or below him?

When the monkey sees the muzzle flash from the hunter’s gun, he lets go of the branch, but since both he and the bullet are subject to the same gravitational acceleration, he falls the same vertical distance as the bullet in the same time, more or less, so he’ll get shot if the gun is aimed directly at him.

Here’s a little game to play Try to shoot the monkey.

The first successful attempt to describe projectile motion quantitatively followed from Galileo’s insight that the horizontal and vertical motions should be considered separately.

Drop two coins of different mass from the same height. Do they hit the ground together? Why?

Push a coin off the side of the table and drop another from the same height. Do they hit the ground at the same time?

Projectile motion can be described by putting these together.

Galileo argued that, if air resistance could be neglected, the horizontal motion was one at constant velocity, the vertical motion was one of constant acceleration, identical to that of an object falling straight down. Put another way, vertical motion is about the equations of motion, horizontal motion is about speed=distance/time, with a=0. We  illustrate this with an example from history.

The English archers won the Battle of Agincourt in 1415 because the arrows’ release speed hence range of the better-made and longer English bows was greater than the bows used by their French enemies. They could probably count on a release speed of 100 feet per second = 3000 cm/s = 30m/s. 

So 30 cos 45 = horizontal speed = 21.2m/s. Using equations of motion with 30sin45 as u, and acceleration  = g, yields t = 2.16s going up, same coming down so total time – 4.3(3)s. So, range =42.4×4.33 = 183m…as long as air resistance is neglected.

Imagine kicking a rugby ball. Suppose the ball leaves the kicker’s boot with a speed of 20ms-1 at 450 to the horizontal. We can use the equations of motion to find out how high it reaches and speed=distance/time to find out how far it travels horizontally, called the Range.

  • The initial velocity, u=20sin 450
  • The final velocity is zero at the maximum height
  • Acceleration  = g
  • Horizontal velocity = constant with time at 20cos450

Plotting a graph of vertical displacement against horizontal distance shows us the trajectory of the rugby ball.

You will see this spreadsheet in class. NB: for MAX range, take-off angle = 450.

kicking a rugby ball

Equations of Motion

WordPress won’t let me post a spreadsheet here. Instead, here’s the first line of it as an image – you’ll have to create it for yourselves after all. Go from t=0 to t=20s.

Calculating v

Here’s what to do. Click on the images to enlarge them and see how the formulae are created.

The time axis goes to the left. When you’ve put in your u and a data, create the formulae as shown to calculate v and s. Try not to wipe the formulae – you’ll have to do them all over again if you do – that’s the unshaded part. Make sure that the spreadsheet copies any changes down the data set. If you need it all, I’ll put it on a flash drive and we’ll upload it on to your machine in class – as long as your machine is clean!

Use the data to produce a v/t graph and a s/t graph as-is, properly formatted with thin lines. You need to experiment with the scatter function. This will look like the ones on the display board. Change u and a, bearing in mind that I may ask you to explain physically what is happening.

Hand in your data plus graphs.

You are going to need a good hard copy of your s/t graph with constant acceleration.

Click here for a Derivation of Equations of Motion from first principles.

Calculating s

The Gold Leaf Electroscope – Electrostatic Measurement

This is a device to measure or compare static charge. It consists of a metal cap with a rod, with a thin gold leaf attached to it, electrically isolated from an earthed case with a black wax plug. When a charged object is brought near to it, the leaf diverges. This one has a scale to measure the angle of divergence. This happens because electrons can move through the metal and either collect at the cap if the object is positively charged, or on the stem and leaf if negatively charged (electrons either repelled or attracted.) In either case, the charges on the leaf, being the same, will repel each other. The closer the charged object is brought to the cap, the greater the divergence. The device effectively behaves like a voltmeter measuring the p d between leaf and case. See here for an animation.