Three short experiments: Measurement uncertainties

Página 1

F1: Horizontal throw:

Goal

The goal of this experiment is to determine the range to which a certain quantity is confined. This will be done by repeatedly measuring the distance that a ball reaches when launched horizontally several times, due to the fact that the landing position of the ball will experience variation over range. After these measurements, the task is to analyze the errors and compare the results to a model in which only uniform gravitational acceleration was used to determine the ball's speed.

Theory

A small ball is placed inside a thin, long tube and launched horizontally. After the launch, the ball's horizontal and vertical speed components can be determined.

1. Horizontal motion with a constnt velocity. The horizontally covered distance by the ball is: x=Vo*t, where Vo is its initial speed and t is the time it takes to reach x.

2. Vertical motion with uniform acceleration due to gravity. Seeing as in this vertical case, the initial velocity is zero, we have: y=1/2*g*t^2, where y is the vertically covered distance, g is the acceleration due to gravity and t is the time it takes to reach y.

If we solve t from the equation for x and substitute the result into the equation for y, then we get an equation of y as a function of x: y=1/2*g*t^2=1/2*g*(x/Vo)^2=g*x^2/2*Vo^2.

If we were to graph y and x using this funtion, then we would get a parabola. But, if we substitute: u=x^2, then we can plot a graph for y versus x^2 and the result is a straight line with a slope of Rc=g/2*Vo^2.

Now, solving for the initial velocity using the equation for the slope, we get: Vo=sqrt(g/2*Rc).

And so, we can determine the relative uncertainty to be: deltaVo/Vo=1/2*deltaRc/Rc.

The uncertainty in the vertical direction is given by deltaY. The uncertainty in the horizontal direction is given by: deltaX^2=2x*deltaX.

However, the uncertainty deltaX should be measured experimentally from the spread in the landing spot of the ball. This can be done by sliding two movable metal slits, which determine the interval in which the ball lands.

Once the measurement uncertainties are determined, we can draw a rectangle around the data points in the graph. Next, we draw two lines from the origin; one that shows the largest possible slope and another that showest the smallest. This way, we can graphically determine deltaRc, which will allow us to calculate deltaVo.

Another way to determine the ball's speed is by using the height. Ignoring friction inside the tube, by the law of conservation of energy, potential energy is converted into kinetic energy: 1/2*m*v^2=m*g*h. If we solve for Vo, we get Vo=sqrt(2*gh).

Measurements

​​​​​​​Start at a small height from the table and launch the ball horizontally from the tube. Repeat several times using increasingly larger heights. Plot the data points in a graph and draw the uncertainty ranges. Draw two possible slopes with the errors to get the uncertainty range of the slope. Using the uncertainty of the slope, calculate the ball's velocity when it leaves the tube and compare that value to what you can calculate from the vertical distance covered by the ball inside the tube.

Página 2

F2: Uniformly accelerated motion:

Goal:

Applying the least square fitting procedure, we can determine the slope of a straight line that describes uniform accelerated motion. From that, the acceleration due to gravity can be calculated.

Theory:

A cart with mass m1 is placed on a horizontal rail. Connected to it by a string and pulley is a cart of mass m2. If this system is released, it will move due to gravity, given by m2*g. There's also a frictional force that resists the motion. According to Newton's law: (m1+m2)*a=m2*g-Ff.

If we were to graph (m1+m2)*a versus m2, we would get a slope that represents g, but that doesn't pass through the origin.

Experimental setup

-Horizontal rail with pulley

-Cart

-Smart timer

-Small weights to vary m2

-Scale to measure the weights

-Spirit level

-Two additional weights to vary m1

Measurements for the acceleration will be performed directly with the smart timer. The pulley has spokes which activate an optical latch when the pulley turns around.

Measurements:

We use the spirit level to make sure the rail is perfectly horizontal. Next, we measure the acceleration of various masses m2 for constant m1. When that's done, make a graph of (m1+m2)*a versus m2, and using the uncertainty ranges, draw two possible slopes for g. Now, determine the values for g and Ff, as well as their uncertainties.

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Página 3

F3: Distance law for the light intensity of an LED:

Goal

We'll check if the distance law holds in this case by measuring the distace dependence of the light intensity of an LED using double-logarithmic paper.

Theory

Because of its small size, we can consider the LED to be a point source that emits light uniformly within a solid angle, as long as the distance to it is large enough. Due to this, the intensity of said light will decrease quadratically with the distance: I is proportional to r^-2, or, I(r)/I(Ro)=(RO/R)^2.

The emitted light will be collected by a photodiode, which will convert it into electrical voltage which is proportional to its light intensity, so: V is proportional to I.