List and comment on the percent errors of the two tuning forks. Which tuning fork had a lower percent error? What could account for error in measurement?

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Analysis Questions

List and comment on the percent errors of the two tuning forks. Which tuning fork had a lower percent error? What could account for error in measurement?

List and comment on the percent errors of the two beat frequencies. Which combinations of tuning forks had a lower present error? What could account for error in measurement?

 

How do we give something a constant angular acceleration? How can we measure this angular acceleration? What are the directions of an object’s velocity and acceleration vectors when it is subject to non-uniform circular motion?

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Florida Institute of Technology ©2020 by J. Gering

Experiment 10 Newton’s Second Law for Rotation

Questions .

How do we give something a constant angular acceleration? How can we measure this angular acceleration? What are the directions of an object’s velocity and acceleration vectors when it is subject to non-uniform circular motion?

Concepts .

When an object moves in a circle and its angular speed (omega, 𝜔) changes, the object experiences an angular acceleration (alpha, 𝛼). In straight-line motion, a net force makes something accelerate. For rotation, a net torque gives an object an angular acceleration. In this experiment, you will apply a constant torque to a circular disk that can rotate on a low friction bearing. The disk will ‘spin up’ to a higher and higher angular speed. You will calculate the disk’s constant, angular acceleration and relate it to the applied torque.

Figure 1. Figure 2.

Figure 3.

!

v 1

!

v 2

Δ !

v / Δt Δ !

v

!

v 1

!

v 2

acen atan

!

a = Δ !

v / Δt( )

!

a

10 – 2

Figure 4.

In Figure 1, a particle moves clockwise in non-uniform circular motion. Its angular and linear speeds increase. The subscripts 1 and 2 refer to early and later times.

Figure 2 shows the change in the particle’s velocity vector. The acceleration vector is just a scaled version of the change in the velocity vector. The scaling factor is the time interval.

Figure 3 shows how the acceleration vector has tangential and centripetal components. The tangential component is

given by , and as usual .

The radius of the circle and the angular acceleration are constant. So, the tangential component of the acceleration remains constant in time. However, due to the speed- squared term, the centripetal component increases with time.

Figure 4 shows both the velocity (blue) and acceleration (red) vectors of the particle at four successive times. The velocity vector is always tangential and grows linearly in length. The acceleration vector also grows in length. It starts out tangential but becomes more and more centripetal.

For a constant angular acceleration, the angular speed, w, obeys a rotational kinematic equation similar to straight-line motion: . If the disk starts from rest, = 0.

The linear and rotational versions of Newton’s Second Law are extremely similar: and . Analogous to the mass m in the linear version, the rotational mass of the object about a given axis of rotation is called the object’s rotational inertia or the moment of inertia, I. Here the word moment is a statistical term that measures how spread out the mass is distributed from the axis of rotation. A dumbbell shaped object is more difficult to rotate than a solid sphere of the same mass. This is because more of the dumbbell’s mass is farther from the middle of the dumbbell, which is typically where the torque is applied. Differently shaped objects have different rotational inertias. Also, the same object has different moments of inertia depending on the choice of the rotation axis. We will use a solid disk of mass M and radius R with an axis perpendicular to the plane of the disk and through the disk’s center. In this case, the moment of inertia is I = ½MR2 .

!

a

!

v

a tan = r

α acen = v 2

r

ω(t) =

ω 0 +

αt ω 0

!

Fnet = m !

a !

τ net = I !

α

10 – 3

A low-detail diagram of the apparatus is shown in Figure 5. A hanging mass falls and accelerates the disk by unwinding a string wound around one of three small (stepped) pulleys mounted to the disk. A photogate is used to measure the angular speed of the vertical pulley. From this you can use Logger Pro to calculate the linear speed of the string and its acceleration. Let us take as the linear speed of the string; the sub-script V stands for the vertically mounted pulley and S for the stepped pulley. Then the angular speed of each pulley is written as follows. Each r is the radius of each pulley.

and (4)

The tension in the string is the force that exerts the torque. However, this tension is not simply the weight of the hanging mass. We will ignore the mass of the vertical pulley and all friction. Then Newton’s Second Law gives for the falling weight:

Ignoring the mass of the vertical pulley allows us to write . (5)

Applying the definition of torque to the disk and using the rotational version of N2 gives:

Substituting for the tension gives: .

Substituting from Eqn. (5) and solving for the linear acceleration gives

(6)

Method .

Figure 5 depicts the second part of this week’s apparatus: a heavy disk that can rotate in a horizontal plane on a low friction bearing. A two-claw clamp and a support rod for a pulley have been omitted for simplicity. The acceleration of the string is measured by a photogate mounted around the pulley and connected to the Lab Pro interface. A similar arrangement was used in the Newton’s Second Law experiment.

v

ωV = v

rV

ω S = v

rS

T − mg = −ma

a =

αrS

τ ≡ rS

T = I

α

τ = rS mg − m

αrS( ) = I

α

a = mgrS

2

I + mrS

2

Figure 5. The Pasco Introductory Rotation Apparatus.

Procedure .

1) Use a Vernier caliper to measure the diameter of whichever stepped pulley you want to use. Divide the diameter by two to find r S .

2) Measure the mass M of the plastic disk. The disk’s mass is greater than 610 grams, which is the maximum mass the triple beam balance can measure. However, hanging a tare weight on a small metal stud found at the end of the scale extends the balance’s range. These are called tare weights. One tare weight adds 500 grams to the reading. In reality, this mass is 147.5 grams but its torque on the apparatus yields a 500 gram equivalent mass. Using the same proportions, a 200-gram hooked mass can be used as a tare weight. This will add 678 grams to the scale reading.

3) Measure the radius, R, of the plastic disk. We will ignore the slight difference in the moment of inertia formula that would result from including the shape of the stepped pulley. Calculate the moment of inertia of the disk using I = ½MR2 .

4) Set-up the apparatus as in Fig. 5. Use white, kite string or fishing line as the string. Do not tie the string to the screw at the top of the stepped pulley; just wrap it around the screw a few times. This way you can easily remove the string. Then thread it through

the holes drilled in the pulley and then wrap the string around the chosen pulley. For

example, if you use the largest diameter pulley, the string should go through three holes.

5) Setup the software for using the photogate and the 10-spoke pulley by executing these menu commands: Experiment > Set Up Sensors > Show All Interfaces >  > Set Distance or Length… > Ultra Pulley 10 Spoke In Groove.

6) Also increase the sampling rate to 50 samples / sec. Execute these commands: Experiment > Data Collection.

7) Adjust the length of the string so the weight hanger does not strike the floor when all of the string has unwound. Place a foam pad beneath the hanging weight to avoid damage if

stepped pulley

rotating disk

hanging

mass

vertically

mounted

pulley

10 – 5 the string breaks. Use slotted masses and the Beck mass hangers. The plastic 5-gram mass hangers are fragile and can break too easily.

8) Lower a mass before recording data. Examine the apparatus to make sure the disk accelerates, and that friction is minimal. Use Logger Pro to measure the linear acceleration of the 10-spoke pulley for five different falling masses. Do not use more than 150 grams of hanging mass.

9) Use Logger Pro to acquire a velocity vs. time graph for when the hanging weight falls. Use the slope of a tangent line feature to obtain five accelerations for one run. Record your data in a spreadsheet. Later, calculate the average and sample standard deviation for the five accelerations.

10) Change the amount of hanging mass and perform additional trials.

11) For each trial, calculate the linear acceleration from Eqn. (6). Compare it to the measured value. Enter these theoretical accelerations into the spreadsheet.

12) Calculate the differences between all experimental and theoretical accelerations. Compare each difference to the standard deviation in each experimental value. If the standard deviation is as large or larger than the difference, then the results agree to within experimental error. There may be no need to propagate errors through Eqn. 6.

13) Comment on sources and types of errors in your discussion.

Change the mass of object 2 as shown in the table 2 and for each mass record the gravitational force between the two objects in table 2.

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Worksheet Using Phet Interactive Simulation

Dep. Of Applied Physics and Astronomy University of Sharjah

This activity consists of two Parts

  • Part one: Gravitational force versus distance.
  • Part two: Gravitational forces versus mass.

To be familiar with the Gravitational force magnitude direction and the parameters affect this force using Phet simulation open the following link and play with it.

https://phet.colorado.edu/sims/html/gravity-force-lab-basics/latest/gravity-force-lab-basics_en.html

Objectives:

⦁ Satisfy Universal Gravitational law experimentally

⦁ Study the parameters that affect the Gravitational force. (distance and mass)

3- Find experimentally the Gravitational constant G.

Theoretical Background:

Universal Gravitational Law: “The magnitude of the Gravitational force that an object exerts on another is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.” Mathematically, the magnitude of this force FG acting on two objects (m1, m2) is expressed as:

FG=G

Where r is the distance between the objects and G is a constant of proportionality, called the universal gravitational constant, G = 6.67 × 1011 Nm2/Kg2.

Part one:

To satisfy the objectives do the following steps.

⦁ Click on the following link and control the masses of the two objects m1 at 4x109kg and m2 at 6x109kg, record your data in table 1.

https://phet.colorado.edu/sims/html/gravity-force-lab-basics/latest/gravity-force-lab-basics_en.html

⦁ Change the distance between the two masses as in shown in the table 1.

⦁ Record the force value for each distance.

⦁ Fill table 1 by finding r2 and 1/r2.

Table 1

m1= …………. m2=………….

r (km) r2 (km)2 1/r2 (1/km)2 FG (N)

9

8

7

6

5

4

3

Part two:

To satisfy the objectives do the following steps.

⦁ Click on the following link and adjust the masses m1 at 2×109 kg and the distance between the two objects r at 5km, write their values in the table 1.

https://phet.colorado.edu/sims/html/gravity-force-lab-basics/latest/gravity-force-lab-basics_en.html

⦁ Change the mass of object 2 as shown in the table 2 and for each mass record the gravitational force between the two objects in table 2.

Table 2

m1= …………. r = ………….

m2 (kg) FG (N)

10×109

9×109

8×109

7×109

6×109

5×109

4×109

3×109

Data Analysis

Part one:

⦁ Uses excel software and plot a graph relates FG and r. comments on the graph.

⦁ Uses excel and plot one more graph relates FG and 1/r2. Use the graph to find the universal gravitational constant G.

⦁ Calculate the percentage error in G (Gknown=6.67 × 1011 Nm2/Kg2)

Note: Attach the graphs to your sheet

Part two:

⦁ Uses excel software and plot a graph relates FG and m2. Comments on the graph.

⦁ Use the graph to find the universal gravitational constant G.

⦁ Calculate the percentage error in k (kknown=6.67 × 1011 Nm2/Kg2)

Note: Attach the graphs to your sheet

 

What are the forces? What is the displacement? Use Hooke’s Law to determine what the displacement should be. Does your prediction match what the software shows?

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Energy

Go to https://phet.colorado.edu/en/simulation/hookes-law

Part 1:

⦁ Set the spring constant to 200N/m. Click in all of the check boxes to show all vectors and values. Apply a force of 60 N by dragging the red plunger or the red slider.

What are the forces?

What is the displacement?

Use Hooke’s Law to determine what the displacement should be. Does your prediction match what the software shows?

⦁ Apply a force of -50 N by dragging the red plunger or the red slider.

What are the forces?

What is the displacement?

Use Hooke’s Law to determine what the displacement should be. Does your prediction match what the software shows?

Part 2: At the bottom of the screen, click the icon that says Energy. If you don’t see it, return to

https://phet.colorado.edu/en/simulation/hookes-law and click on Energy

Set the Spring Constant to 200N/m. Set the displacement to 0.2 m.

What is the applied force?

Use Hooke’s Law to determine what the applied force should be. Does your prediction match what the software shows?

Turn on the Energy Plot. What is the value of the Spring Potential Energy? Use to determine what the energy should be. Does your prediction match what the software shows?

Turn on the Force Plot, and check the box to show the Energy. Recall that . So the work required to stretch the spring is the shaded area under the Force graph.

Determine that area using geometry or calculus.

What is the slope of the Force graph? What does that value represent?

 

Apply the Legendre transformation to the Entropy S as a function of E, specifically for the ideal gas according to the Sackur-Tetrode equation: S(E, = KBN(2 mkN) -AK/ + 2 k277r3h6/ e5m3

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Legendre-transformed

The Legendre transform g of a function f is:

g(m) = f(x(m))- m • x(m) mit

x(m) = (r)-1(m).

Show that h(y) := f(-y) is f Legendre-transformed twice. This demonstrates that no information is lost during the Legendre trans-formation.

Apply the Legendre transformation to the Entropy S as a function of E, specifically for the ideal gas according to the Sackur-Tetrode equation: S(E, = KBN(2 mkN) -AK/ + 2 k277r3h6/ e5m3 ))

 

The vertical scale is set by Ba = 26.076 μT and Bb = 35.414 μT. What is the radius of the smaller semicircle?

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Radius

The current-carrying wire loop in the figure on the left lies all in one plane and consists of a semicircle of radius 20 cm, a smaller semicircle with the same center, and two radial lengths. The smaller semicircle is rotated out of that plane by angle θ, until it is perpendicular to the plane. The figure on the right gives the magnitude of the net magnetic field at the center of curvature versus angle θ. The vertical scale is set by Ba = 26.076 μT and Bb = 35.414 μT. What is the radius of the smaller semicircle?

Find the speed and direction of the wire’s motion as a function of time, assuming it to be stationary at t = 0. Evaluate for t = 0.5 s. Take positive to the right and negative to the left.

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Direction of the wire’s motion as a function of time,

A metal wire of mass m = 0.400 kg slides without friction on two horizontal rails spaced a distance d = 0.40 m apart, as in the figure. The track lies in a vertical uniform magnetic field B = 0.50 T. There is a constant current i = 0.35 A through generator G, along one rail, across the wire, and back down the other rail.

Find the speed and direction of the wire’s motion as a function of time, assuming it to be stationary at t = 0. Evaluate for t = 0.5 s. Take positive to the right and negative to the left.

 

Are W, M1,M2 still constants of motion? How about M3? What are the physical explanations of your results above? Now consider a similar medium but with linear loss for the ω1 wave only. Do any of W, M1, M2, M3 remain constants of motion? Why?

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Nonlinear Optics

(50 points) The equations of motion describing nonlinear coupling of three plain waves in a lossless χ(2) medium are given in the slowly-varying envelope approximation as where ωj and nj are the angular frequency and refractive index, respectively, of jth wave (j = 1,2,3). K is the effective second-order susceptibility. c is the speed of light in vacuum. ∆k is the phase mismatch per unit length.

(a). What kind of nonlinear processes do those dynamical equations describe? (10 points)

(b). Use those equations of motion to show that the following quantities are constants of motion (30 points)

 

The above equations are known as the Manly-Rowe relations.

(c). Give a physical explanation as to why Manly-Rowe relations hold in lossless media (10).

  1. (50 points) Equations (1)-(3) are for a lossless χ(2) medium. Now consider a χ(2) media with linear propagation loss for the ω3 wave only. The equations of motion then become where Γ is the loss coefficient per unit length.

(a) Are W, M1,M2 still constants of motion? How about M3? Show your mathematical derivations.

(b) What are the physical explanations of your results above?

(c) Now consider a similar medium but with linear loss for the ω1 wave only. Do any of W, M1, M2, M3 remain constants of motion? Why?

Solve each problem clearly and completely, starting from first principles, including all steps and the values of initial parameters. Comment on the error between calculated forces and measured forces, and account for any significant differences.

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Case 1: Traffic Light

A traffic light is hanging from two wires set at known angles. What is the magnitude of the tension in each wire?

Set up this system based on the figure. Hang an object of known mass from the center of a meter-long string with loops on both ends for attaching the spring scale. It may be easier to tie a second shorter string to the center of the first string and tie the object to the short string. Hold the long string in such a way that two different angles, θ1 and θ 2, are formed. It may be helpful to have an assistant, or tie one end of the string to a fixed object.

Record the mass or weight of the object and the two angles on the Report Sheet and solve for the two tensions. You can set the system down while you are doing this, or you can immediately measure the two tensions before doing the calculation. If you measure the tensions last, be sure to re-create the system with the exact same angles.

Carefully attach the spring scale to one end of the string by holding the loop with the hook, making sure that the two angles of the string have not changed. Record the tension, T1, on the Report Sheet. Repeat this procedure for the other end of the string, recording the second tension, T2, on the Report Sheet.

Compare the calculated tensions with the experimental values measured by the spring scale.

Note: This system is only vector addition and does not require the calculation of torques.

Show All

Case 1 Example 1! ”

Show All

Case 1 Example 2! ”

Case 2: Painters on a Scaffold

Two painters with different masses are standing on a scaffold of a certain length and different distances from each end. The cables supporting the scaffold are vertical and attached to each end. What is the tension in each cable?

Set up this system based on the figure using a meter stick or yard stick. Tie two strings about 50 cm long to each end of the stick. The other end of each string should have a loop in it for attaching the spring scale.

Tie or hang two objects with different masses from two points on the stick. The locations should be chosen so that the system is not symmetrical (not the same distance from each end of the stick). Hold the strings so that the stick is horizontal.

Record the distances, masses, and the mass of the stick (usually around 100 grams) on the Report Sheet. Solve for the tension in each string.

You can set the system down while you are doing this, or you can immediately measure the two tensions before doing the calculation. In either case, when measuring the tensions make sure the stick is horizontal.

Carefully attach the spring scale to the loop of one string by holding the loop with the hook, making sure that the spring scale is vertical and the stick remains horizontal.

Record the tension, T 1, on the Report Sheet. Repeat this procedure for the other string, recording the second tension, T2, on the Report Sheet. Compare the calculated tensions with the experimental values measured by the spring scale.

Note: all angles between applied forces and the horizontal stick are 90 degrees.

Show All

Case 2 Example 1! ”

Show All

Case 2 Example 2! ”

Case 3: The Angled Boom

A crate is hanging from a boom set at a certain angle and held in place by a horizontal cable. What is the tension in the cable?

Set up this system based on the figure. Tie a string about 50 cm long (the ‘cable’) from a point on a meter stick around 5 cm from the end. The other end of each string should have a loop in it for attaching the spring scale. Hang a mass from a short string tied to some point on a meter stick (the green block in the figure). Anchor the pivot point of the stick (the lower end on the left side of the figure) to a flat surface. Hold the longer string horizontally and choose an angle θ. Angles of 30, 60 or 45 degrees can make the calculations easier.

Record the distances to the two attachment points from the pivot point, as well as the mass of the hanging object and the angle of the stick on the Report Sheet. Solve for the tension in the horizontal string.

You can set the system down while you are doing this, or you can immediately measure the tension in the horizontal string before doing the calculation. In either case, when measuring the tension make sure the string is horizontal and the angle of the stick is the same as in the original position.

Attach the spring scale to the loop in the horizontal string and record the tension on the Report Sheet. Compare the calculated tension with the experimental value measured by the spring scale.

Note: Choose the pivot point at the base as the center of torque when calculating the tension in the string.

Show All

Case 3 Example1! ”

Show All

Case 3 Example 2! “

Conclusions

Solve each problem clearly and completely, starting from first principles, including all steps and the values of initial parameters. Comment on the error between calculated forces and measured forces, and account for any significant differences.

 

What is a force that moves an object toward another object?

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Force

What is a force that moves an object toward another object?