PHYS 111LV lab 7
Introduction: According to the syllabus,this is when we do torque and angular momentum.
However,such labs are difficult with a simple labkit.So instead,we will look at a pendulum.
We will not totally ignore toque,however,but instead will use it in the derivation of the period of a pendulum.
An ideal pendulum consists of a massless string or rod and a pendulum bob,which is a massive object at the end of the string or rod.The ideal bob is massive,but has a small volume and can be regarded as a point-mass.
The equation for the period of a pendulum can be found as follows:consider the suspension point of the pendulum as the center or rotation of the bob.Then the radius that the bob has from its center of rotation is L,the length of the pendulum.The moment of inertia of a point mass is
I=mL² .
When the pendulum makes an angle θ with respect to the vertical,the torque due to gravity on the bob is
t=-mg L sinθ,
Where the negative sign comes from the fact that if we measure the angleθ as positive to the right,then the torque will induce a clockwise rotation which by convention in PHYS 111 is a negative torque.Using
α=a/L,
Where a is the angular accelerationof the bob,and a is the linearaceleration,we apply Newton's second law for rotations:
lα=Zt
Which gives us
mL²a/L=mLa=-mgLsinθ .
Now,the equation mLa=-mgLsinθ can be simplified.We notice that for small angles,the movement of the bob in the y-direction is very small compared to its movement in the x- direction.Hence,the acceleration can be written as
a=ax
Lsinθ is x,so the equation for motion will be mLax=-mgx.Dividing by mL,we get ax=-(g/L)x.
This equation is to be compared to that for a mass on a spring.Newton's second law in this case becomes
ax=-(k/m)x,
where k is the spring constant,m is the mass on the end of the spring,and xis measured from the equilibrium position.The period of oscillation forthe mass at the end of the spring is
P=2π√m/k
Comparing the acceleration of the mass on the spring with that of the pendulum bob,we see that if we replace k/mwith g/L,the accelerations are identical,indicating (and think about this a little)
P=2π√L/g
is the period of the pendulum.
Our main assumption in findingthis period is that the angle the pendulumswingsover is
“small”,which means that if you were to measure the maximum angle of the pendulum from vertical,that this angle in radians should be much less than 1.A good rule of thumb is under about 20-degrees.
Description of Experiment: In this experiment use the lead fishing weight from your labpaq,or some similar weight as a pendulum bob.Use the fishing line attached to this weight to create a pendulum.Find some way to suspend the pendulum (tape on the top of a doorjamb works OK).Measure the period of the pendulum using a stopwatch.Try measuring the time for 10 periods.(A period is a complete cycle of motion).If the uncertainty in your stopwatch is about 0.2 s,then this would be the uncertainty in 10 periods,meaning the uncertainty in the single period would then be 0.02 s.
Try this with 5 different lengths of pendulum.
For each pendulum length,find the square or the period,P².Forthe five different pendulum lengths,plot P²vs L,that is plot a point with P²as the y-position and the length L as the x-position.Draw a line of best fit through these points,with error bars.Find the slope of the line of best fit,and an uncertainty in the slope of the line of bestfit (like you did in lab 2). Now according to one of the equations in the introduction,
P²=4π²L/g
So the slope of your line of best fit should be related to g,the aceleration due to gravity.What do you measure g to be,with uncertainty?
Is this consistent with the value of g you found in lab 1(use the z'test to test consistency). Please discuss anyissues,and if you had to measure g,whichmethod would you use,this pendulum method,or the method from lab 1?
No questions this week.