PHYS261 Cumulative Lab
Using the vernier calipers, estimate the uncertainty in a measurement of the length of a side of one of the blocks
+-one of smallest units (+-0.1 mm)
The small angle approximation says that sin(theta)=theta (THETA MUST BE IN RADIANS). Find the percentage error (sin(theta)-theta)/theta this approximation introduces for an angle of 0.020 radians.
-0.000067
Using the ruler, how precisely can you measure the length of the pendulum?
0.5 mm or 0.25 mm
A cart of mass m=0.25 kg rides on an air track that is tilted 3 degrees from the horizontal. What is the magnitude of the cart's acceleration?
0.51 m/s^2 9.8*sin(theta)
A sonic ranger measures positions X=0.10 m, 0.15 m, 0.22 m, 0.28 m at times t=0 s, 0.10 s, 0.20 s, and 0.30 s. Find the magnitude of the average velocity between t=0.10 s and 0.20 s.
0.7 m/s Deltax/deltat
Find the theoretical value for the rotation period for a pendulum bob with height y=0.070 m, length L=0.20 m, y0=0.020 m, and extension arm length r0=0.020 m. Use g=9.801 m/s2.
0.834 s Period (tau)=2*pi*sqrt((r*sqrt(L^2-(r-r0)^2))/(g*(r-r0))) r=r0+L*sin(theta)
Suppose that the temperature of the bulb in the Charles' law apparatus increases from 0.0 to 100.0 degrees C. By what multiplicative factor will the pressure increase by?
1.366 PV=nRT P2/P1=T2/T1 T MUST BE IN KELVINS
If the motor is turning at 3000 rpm and the gear steps this motion down by a factor of 27, what is the frequency of the rotation of the mass in Hz?
1.85 Hz (3000/60)/27
Find theta in degrees when f=1.47 Hz, f0=1.44 Hz, and Q=90.0. The value of theta should be specific to lie within the range of 0 to 180 degrees.
165 degrees Theta=arctan((1/Q)*((f*f0)/(f0^2-f^2)))
Using g=9.801 m/s^2, find the period of a 1.000 m long pendulum.
2.007 s T=2*pi*sqrt(L/g)
Consider the properties of a string that has a length of 1.0 m and a wave speed of 45 m/s. What is the fundamental frequency of the string?
22.5 Hz v=wavelength*frequency
Sketch the shape of the spring when it is excited in the third harmonic.
3 anti nodes and 4 nodes
One radian is equal to how many degrees?
57.3 degrees (180/pi)
A 1.0 cm block is placed under one leg of the air-track. How much time will it take a cart to reach the bottom of the track? Assume the cart starts from rest, the track is 1.5 m long, and the distance covered by the cart is 1.5 m
6.776 s d=vi*t+(a*t^2)/2 a=g*sin(theta)=g*.01/1.5
Yoy measure the lengths of three sides of a block and find a=12.23 mm, b=14.51 mm, c=7.45 mm with an error of +-0.030 mm in each measurement. What is the uncertainty in the volume of the block?
6.8 mm^3 (DeltaV/V)^2=(deltaa/a)^2+(deltab/b)^2+(deltac/c)^2
A mass and spring system has a resonance f0=1.40 Hz with a full-width of f(FW)=0.021 Hz. What is the quality factor Q?
66.67 f(FW)=f0/Q
Consider a 1.5 m long string with linear mass density of 0.0010 kg/m and a tension of 0.50 N. Find the frequency in Hz of the fundamental mode of the string.
7.45 Hz v=sqrt(T/mu) v=wavelength*frequency
Suppose that the volume of the gas is 25 cc and the pressure is 30.0 lbs/in2. If the plunger is depressed so that the volume is decreased to 10 cc, what will the pressure be?
75 lbs/in2 P1V1=P2V2
Suppose the drive frequency f equals the resonance frequency f0. What is the phase difference theta between the position of the drive and the mass?
90 degrees f=0-->theta=0 f=f0-->theta=90 f>>f0-->theta=180
In this lab (lab 7) a periodic force is applied to the cart using:
A spring attached to a speaker
To what uncertainty can the drive frequency be determined in this experiment?
About 0.00033 Hz because it can measure down to 1 mHz
Use the thermometer to accurately measure room temperature and estimate the uncertainty in this measurement.
About 20 degrees Celsius Uncertainty: Using Charles Law Apparatus: deltaT=0.003*T+.02 Using thermometer: deltaT=0.5 or 0.25 of smallest division
Without using a thermometer, how can you establish a temperature of 100 degrees C?
Boil water
A block is measured to have a density p=2.76 g/cm3 with an uncertainty of deltap=0.030 g/cm3. Find chi-squared when the measured density is compared to the accepted density p=2.70 g/cm3.
Chi-squared=4 Chi-squared=sum((xi-xtheory)/deltaxi)^2
Make a sketch which shows how the amplitude depends on the frequency.
Cusp
Using your spreadsheet and the following formula __________, find the theoretical value for the rotation period for the mass when the height of the mass is y=0.07 m, pendulum length L=0.20 m, minimum height y0=0.02 m, the extension arm length r0=0.02 m, and g=9.801 m/s2.
DRAW OUT APPARATUS!! Solve for r in terms of L, y, and y0 Use excel to solve for tau (rotation period)
Suppose that you measure the length of a pendulum to be L=1.200 m with an uncertainty of deltaL=0.0020 m. If you use this length and a measured value of the period to find g, what is the uncertainty in g produced by this uncertainty in L? Assume that the measured period T has zero uncertainty.
Deltag=0.016 m/s^2 Deltag=g*sqrt((deltaL/L)^2+(2*deltaT/T)^2)
In this lab, the position of a cart will be measured using a sonic ranger. This measures the position by:
Emitting an ultrasonic pulse
To exit a lab template:
Enter name and section number and save file using macro
The table below lists the position versus time for a cart moving on a tilted air-track. Use the spreadsheet to calculate and plot the velocity versus time and acceleration versus time.
Find average velocity and average acceleration (will have one fewer velocities than positions and one fewer accelerations than velocities). Use excel to plot.
Measure the resonant frequency of this system.
Find period (time between two maximums), then take inverse.
$A$5 means?
Fixed cell
To use a macro, you may have to first:
Go to view tab and click on the macro button
Why is it important to keep the angle small in this experiment? Explain.
If the angle is small, the small angle approximation allows for tau=-mgLtheta
If you make a pendulum 3 times longer, then its period of oscillation will:
Increase by a factor of sqrt(3) T=2*pi*sqrt(L/g)
Solver
Minimizes or maximizes a cell by varying other cells
Make a sketch that shows how the period of the rotating mass varies with the radius of the masses orbit.
Near r=0, tau approaches infinity. Quickly levels off and plateaus, then decreases quickly again approaching negative infinity.
Two carts approach each other, collide, and stick together after the collision. This is an example of what type of collision:
Perfectly inelastic collision
Using the Charles' Law apparatus, a student finds pressures of 760, 840, 950 mm of Hg at temperatures of 26, 57, and 100 degrees Celsius. Use the spreadsheet to make a plot of P versus T and find the slope of the resulting straight line. Does the data agree with Charles' Law? Briefly explain.
Plot in excel. Linear relationship-->P=C*T
Explain how you can use the Charles' Law apparatus to determine the absolute zero temperature.
Plotting temperature vs pressure yields a linear line. The y intercept of the line would be the slope multiplied by absolute zero.
Explain how you could measure the Q of the resonance in this system.
Record the oscillations, find f0 by finding the period and taking the inverse (1/period), find the damping time (td, time where amplitude is 1/3 max amplitude), then plug into equation: Q=2*pi*f0*td/2
Suppose that you wanted to decrease the resonance frequency of the system, what could you do?
Reduce the mass
Suppose that you wanted to reduce the resonant frequency of the system, what could you do?
Reduce the mass Tosc=fosc=(2*pi)/sqrt((2*k/m)-(b/(2*m))^2)
How to plot cartesian coordinates
Scatter chart
Using the Boyle's Law apparatus, a student finds volumes of 40, 30, 20, and 15 cm3 at pressures of 7.5, 10, 15, and 20 lb/in^2. Use the spreadsheet to make a log log plot of P versus V and find the slope of the resulting straight line. Does the data agree with Boyle's Law? Briefly explain.
Slope=-1 So exponential equation has exponent of -1 which is the same as 1/(base of exponent). This correlates to V'=Cb(1/Pavg)-V0 V as function of P: V=(nRT)/P
Make a sketch which shows how the phase depends on the frequency.
Slope=0, then sharp incline, then slope=0
Is a calibration error in the timer (pendulum and g experiment) a systematic or random error in determining g? Can you overcome this by making more measurements with the same timer?
Systematic error. No, the best way to find and overcome a systematic error is to compare your results with results from a completely different apparatus.
Starting at 20.0 degrees C, to what temperature would you have to raise the Charles law apparatus to double the pressure?
T2=313.15 degrees C T2=(P2/P1)*T1 T MUST BE IN KELVINS
What is room temperature on the Celsius scale approximately?
T=20 degrees C
A student measures wave speeds of 5 m/s, 7 m/s, 10 m/s, and 15 m/s for tensions of 1 N, 2N, 4 N, and 9 N. Use the spreadsheet to make a log log plot of the wave speed versus the tension. Also find the slope of the resulting line.
Take log of both variables. Plot. Add trendline with equation.
What provides the vertical force to balance the force of gravity on the pendulum bob?
Tension in the string
Suppose that you wanted to increase the quality factor Q of the system, what could you do?
The mass could be increased.
What does absolute zero temperature mean?
The temperature at which the pressure of an ideal gas would be zero (0 Kelvins)
Two carts collide and bounce off each other. If there are no external forces acting on the system:
The total kinetic energy can change during the collision but the total momentum of the carts must be the same before and after the collision.
What effect does the weight of the bob have in this experiment?
The weight provides a restoring force on the pendulum. It increases tension but does not affect frequency, period, or length. The bob is pulled down by the same gravitational force, no matter what mass the bob has. Thus, motion is independent of the mass.
How can it happen that when two carts collide in this experiment that the total momentum of the two carts is not conserved?
There could be external forces working on the system.
Experiment 6: The Pendulum and g
To determine the dependence of the oscillation period of a pendulum on the pendulum's length and mass. Measurements are used to determine g, the acceleration due to gravity
Experiment 3: X, V, and A Purpose:
To determine the value of g from measured values of X and determine if it agrees with the theory value of 9.801
Experiment 7: Forced Harmonic Motion
To examine the amplitude, resonant frequency, damping time, quality factor, and phase shift of a driven mass-and-spring oscillator
Experiment 9: Ideal Gas Law and Absolute Zero
To examine the relationship between pressure, volume, and temperature of air in a closed chamber. Tested under constant temperature (Boyle's Law) and constant volume (Charles' Law). Also finds Absolute Zero, the temperature at which the pressure of an ideal gas would be zero.
Experiment 2: errors Purpose:
To learn how to estimate uncertainty in measurements, how to do error propagation, and why it is important to understand the uncertainty in measurements
How is the spacing between nodes of a standing wave related to the wavelength?
Wavelength=2*spacing between two nodes
Explain how the optical picket fence works and how this system can find the velocity of the cart.
When the cart moves through an optical gate, the dark parts of the picket fence block the sensor beam. The gates keep track of when the light intensity changed and the use the know distance between the pickets to find the speed of the cart.
Suppose you measure g=9.7 m/s^2 with an uncertainty of deltag=0.060 m/s^2. Is this consistent with the accepted value g=9.801 m/s^2?
Yes Chi-squared=sum((xi-xtheory)/deltax)^2 Chidist(2.83,1)=0.09 USE EXCEL FOR CHIDIST
The function that is plotted as a line on the log-log graph to the right is a power-law function of the form y=a*x^n. From this plot, what is the value of a?
a=y-intercept
A cart starts at X=0.5 m with an initial velocity of 1 m/s heading up the airtrack. Assuming that the airtrack is tilted at 0.01 radians, use the spreadsheet to calculate and plot the position versus time and the velocity versus time of the cart.
d=d0+v0*t+(a0*t^2)/2 a0=g*sin(theta) v=vi+a0*t
The function that is plotted as a line on the log-log graph to the right is a power-law function of the form y=a*x^n. From this plot, what is the value of n?
n=(log(y2)-log(y1))/(log(x2)-log(x1))
Suppose there are 100.0 cm3 of nitrogen at a temperature of 0.0 degrees Celsius and a pressure of 1.500*10^5 Pa. How many miles of nitrogen are there?
n=0.0066 PV=nRT R=8.31 if Pa and m^3 R=0.082 if atm and L
A student measures the four lowest resonant frequencies of the string as 12 Hz, 25 Hz, 35 Hz, and 50.0 Hz, with node-to-node spacing of 2.0 m, 1.0 m, 0.67 m, and 0.50 m. Calculate the wave velocity for each of the four different modes.
v1=48 m/s v2=50 m/s v3=46.9 m/s v4=50 m/s v=wavelength*frequency
How is the velocity of the wave related to the mass hanging off the end?
v=sqrt((mg)/mu)
What is the relationship between the frequency of a wave and its wavelength?
v=wavelength*frequency
Find x0 when A0=1.00 mm, f=1.40 Hz, f0=1.45 Hz, and Q=80.0.
x0=7.26 mm x0=(A0/2)/sqrt((f^2/(Q^2*f0^2))+(f^2/f0^2-1)^2) KEEP IN MILLIMETERS
Experiment 5: Centripetal Motion
To measure an object undergoing centripetal acceleration
Experiment 4: Momentum and Drag
To observe the drag force and conservation of momentum
Using the ruler, and without using the vernier calipers, estimate how precisely you can measure the length of one of the sides of one of the blocks.
To within 0.5 mm or 0.25 mm
A cart with mass 0.3 kg and velocity 0.1 m/s collides on an air track with a cart with mass 0.4 kg and velocity -0.2 m/s. If the final velocity of the first cart is -0.1 m/s after the collision, what is the velocity of the second cart?
-0.05 m/s mv=mv
A cart with mass 0.30 kg and velocity 0.10 m/s collided on an air-track with a cart with mass 0.40 kg and velocity -0.20 m/s. What is the final velocity if the two carts stick together?
-0.0714 m/s m1v1+m2v2=(m1+m2)vf
How precisely can you measure temperature in the Charles' Law apparatus? How precisely can you measure the pressure?
.01 degrees Celsius .01 kPa DeltaT=0.003*T+0.2 DeltaP=Pavg/500
If a 1 cm block is placed under one leg of the air track, what is the tilt angle of the air track?
0.006667 radians asin(0.01/1.5)
A block is measured to have mass M=25.3 g and volume V=9.16 cm3 with an uncertainty of deltaM=0.050 g in the mass and deltaV=0.050 cm3 in the volume. What is the uncertainty in the density?
0.016 g/cm3 (Deltap/p)^2=(deltaM/M)^2+(deltaV/V)^2
A cart with mass 0.30 kg and velocity 0.20 m/s collided on an air-track with a cart with mass 0.40 kg and velocity -0.10 m/s. What is the final velocity in m/s of the two carts if they stick together?
0.029 m/s m1v1+m2v2=(m1+m2)vf
Suppose you use the scale to measure the mass of the brass block. What is the uncertainty in the mass?
0.033 g
You measure the length of the same side of a block five times and each measurement has an uncertainty of 0.10 mm. What is the uncertainty in the best estimate for b?
0.0447 =0.10/sqrt(5)
A cart of mass m=0.20 kg rides on an air track that is tilted at an angle 2 degrees from the horizontal. What is the magnitude of the total force on the cart?
0.068 N Mass*9.8*sin(theta)
Suppose the air-track is tilted at an angle of 0.01 radians from level, what will be the acceleration of a friction-less cart?
0.097998 m/s2 a=g*sin(theta)
You measure the length of the same side of a block five times and find: b=11.0, 11.1, 11.2, 11.2, and 11.3 mm. What is the best estimate for b?
11.16 mm Take the average
Suppose a pendulum bob is swinging with an amplitude of 0.1 radians and a frequency of 2*pi radians/sec. The angle the pendulum's string makes with the vertical direction can be written as angle=amplitude*cos(frequency*t). Use the spreadsheet to compute angle at times 0, 0.1, 0.2, ..., 1.9, 2.0 seconds and then plot angle vs time
Plug and chug Excel Use given equation
Experiment 8: Waves on a String
To examine waves on a string. To find the resonance frequency of the fundamental standing wave, and its harmonics, and determine the propagation speed of the waves by measuring the wavelength and frequency
Measure the time for 20 swings of the pendulum. What is the period?
Use Logitech. Period=time for one swing
What is the maximum height y that the pendulum can reach in this experiment?
y0+L (From surface to the bottom of the pendulum plus the length of the pendulum
