Mechanics Tutorial 7 - Free Fall and Terminal Speed
Contents |
Action of Gravity Fields
Gravity is an attractive force between two objects with mass. It is very weak. The only reason we feel it at all is because the Earth is a very big object with a mass of 5.98 × 10^{24} kg (which is heavy). Gravity is always attractive. Gravity causes objects with a mass to have a weight.
1 kg weighs 9.81 N on the Earth and 1.6 N on the
Moon
The
acceleration due to gravity on the
Earth is 9.81 m s^{-2}.
It is given the physics code
g
and
the value is sometimes approximated to 10 m/s^{2}.
(The latter approximation is perfectly acceptable in GCSE
examinations, but you will lose marks if you use it at A-level.)
1 kg weighs 9.81 N on the Earth. Therefore we can also write the value of
g
as 9.81 N kg^{-1}.
Weight is a force measured in Newton (N). It is not
measured in kilograms. It is depressing how many students make this
mistake. |
The acceleration near the Earth’s surface remains
constant. An object like a ball thrown vertically upwards is always
accelerating towards the Earth at 9.81 m s^{-2}. The split second
it leaves the hand, it is accelerating downwards at 9.81 m s^{-2},
regardless as to whether the ball is going up or down.
By convention:
Upwards is positive;
Downwards is negative.
Consider a ball being thrown up vertically. The displacement time graph look like this:
We can deduce the following from the graph:
The ball slows down under a constant acceleration of -9.81 m s^{-2} (downwards).
At the top its velocity is zero, but it is still accelerating at -9.81 m s^{-2} (downwards)
Then it falls downwards at an acceleration of
-9.81 m s^{-2}.
We are ignoring air resistance. The graph is a parabola. It is very important to take into account of the signs, especially when a ball is being thrown up in the air and then is dropping down again.
The corresponding velocity-time graph is like this:
The acceleration is constant and negative, so gives a negative gradient. This will have the value of g. Mathematically we can write:
This gives rise to the equation:
If the ball is caught at the same height, the positive area under the graph is the same as the negative area, giving a displacement of zero. If the ball drops to the ground, the negative area is slightly bigger than the positive area.
The measurement of g by free fall can be done using light-gates and data-loggers, or by timing the drop of a steel ball bearing. This is an assessed practical that you will do with your tutor. You will use apparatus like this:
The procedure is this:
The ball bearing is released when the electromagnet is turned off. It falls through each light gate.
The data-logger needs to be set to time from A to B.
You need to vary
h,
the distance between the light gates.
All objects will accelerate downwards at (-)9.81 m s^{-2}, regardless of mass.
Over a short distance, a small light object will hit the ground at the same time as a larger and heavier object.
Over a longer distance, air resistance becomes important. The air resistance reduces the acceleration. The faster the object, the greater the air resistance. Eventually the air resistance balances the weight, and there is zero acceleration. We have terminal velocity or terminal speed. Since we are not considering direction, we should call it terminal speed.
For a feather, which has a large surface area compared with its mass, the terminal speed is about 10 cm s^{-1}. For a sky-diver, it is about 60 m s^{-1}. For the sky-diver when the parachute opens, it is about 5 m s^{-1}. The value of the terminal speed depends on not only the mass of the object, but also its surface area.
The concept of terminal velocity can be applied to an object falling in any fluid, i.e. liquid or gas. It does not apply, of course, to solids. Nor does it apply in a vacuum. If you throw an object from an space-craft orbiting the Moon, the object will continue to accelerate at a rate of 1.6 m s^{-2} until it hits the Moon's surface. It might be travelling quite fast.
An object falls vertically from a space capsule that is 10 000 m above the Moon's surface. (a) Calculate the speed at which the object hits the surface of the moon. Use g = 1.6 m s^{-2}. (b) Calculate the time it takes to fall. |
Free Fall in Air
Think about a sky divers jumping from a plane:
The animation will help you understand this:
The drag is in the opposite direction to the weight, so the acceleration decreases.
Then the drag (upwards force)
balances the downward force (weight)
Therefore there is no acceleration and the speed is constant. This is terminal speed.
Sketch a speed time graph of a parachutist jumping from an aeroplane, reaching terminal velocity, then opening her parachute. Explain what is happening at each stage. |
Terminal Speed in Liquids (Required Practical)
Dropping objects off tall buildings or lobbing things out of aeroplanes results in certain Health and Safety management issues. It is rather safer to measure terminal speed in a liquid, since the terminal speed is rather lower. And it's easier to time as well. Here is a typical experiment you will do:
When a ball bearing is dropped into a viscous liquid, it almost immediately reaches its terminal speed, and measuring it is simply a matter of timing the motion between two fixed points a known distance apart. You will also have to:
Note the mass, m, of each ball.
Use a micrometer to measure the diameter,
d.
A viscous material is gooey. Runny materials have low viscosity. There are various ways of measuring viscosity, the most common of which is to drop a ball-bearing into the liquid, and measuring its terminal speed. Remember that at terminal speed, the upwards forces of upthrust and drag and the downwards force of the weight are balanced.
The upthrust is the same as the weight of fluid displaced by Archimedes' principle. Therefore, if the weight is greater than the upthrust, the object will accelerate downwards until the drag balances the difference between the weight and the upthrust. This is true of all fluids, for example air, or water, or chocolate.
We can calculate the drag force by Stokes' Law, which we will look at in the optional topic, Turning Points in Physics. The measurement of viscosity is important for the manufacturers of confectionery and lubricating oils. Measurement of terminal speed is one technique.
A small ball bearing is dropped into a cylinder of viscous liquid and allowed to fall until it reaches the bottom. Describe in as much detail as you can the forces that act on the ball bearing, and explain why it falls at a constant velocity. |
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(a) Describe how you would measure the terminal speed of a small ball bearing in liquids of different viscosities. (b) Suggest how the data could be used to determine viscosity. |
A more detailed treatment of the theory of this experiment can be found in Materials Tutorial 4.