# Rotational Equilibrium Lab Conclusion Essay

Torque indicates a force’s ability to rotate an object about an axis. The magnitude of torque about an axis, through point O, is defined by $$\tau = rF sin(\theta)$$ (1)

The distance between the force and the axis, O, is r. θ is the angle between the force vector and r (see Figure 1). Torques that rotate a body counterclockwise about an axis are considered positive, and torques that rotate a body clockwise about an axis are considered negative.

An object in static equilibrium is not accelerated linearly or rotationally. These two conditions of static equilibrium are:

i) $\sum F_{x} = 0$, $\sum F_{y} = 0$ (Vector sum of the forces is zero - no linear acceleration)

and

ii)$\sum \tau = 0$ (Vector sum of the torques is zero - no rotation)

## Procedure:

Concept Checkpoint 1:

1. Discuss with your partner the situation of a see-saw that has a small child and a large child in balance so the beam is level. Draw a diagram with big and small “m”s for the different masses and big and small “d” for the two distances. Define the point of rotation with a star.

2. What direction will each torque be in?

3. Write an equation using your variables to show this state of torque equilibrium.

4. Call over a TA or instructor and explain your conclusion and diagram to them.

### Part A: Model of the Forearm

1. Open Capstone and select Hardware Setup.

2. Connect the force sensor into analog channel A of the Signal Interface. Make sure the switch on the force sensor is set to ± 10 N.

3. Add the Force Sensor to the Interface

4. Set the Sample Rate to 1000 Hz.

5. Calibrate the force sensor using zero weight for the first calibration point and 4.9 N (500 g hanging directly from the sensor) for the second calibration point.

1. Make sure the force sensor and weight are hanging vertically downwards.

6. Drag Digits to the Main Screen.

1. Adjust the number of digits to Increase Precision.

7. Record the mass and weight of the wooden part of the “forearm”.

1. If the forearm is already set up, you can measure the mass later.
8. Torques will be determined about the elbow joint, so distances will be measured from the elbow screw to where each of the three forces are applied.

1. Measure the distance between the forearm's elbow joint and the string tied to the forearm. Record this distance in your e-journal as dtension.

2. Measure the distance from the elbow joint to the center of gravity of the forearm. The center of gravity should be the exact center of the forearm. Record this distance in your e-journal as darm.

3. Measure the distance from the elbow joint to the point at the end of the forearm where the load attaches. Record this distance in your e-journal as dload.

### Setup 1

1. Using the string attached to the force sensor, adjust the wooden arm so that it is at a right angle to the metal rod and parallel to the floor. (Figure 3)

2. Adjust the force sensor until it points in the same direction as the string.

3. Hang a 200 g mass from the string at the end of the wooden arm and calculate the weight.

4. Make sure the wooden arm is horizontal.

5. Use a protractor to measure and record the angle θ.

1. θ is the angle between the string and the wooden arm. It should be smaller than 90 degrees.

6. In Capstone, click Record and record the tension, T, from the digits display.

7. Compute the net torque about the elbow joint.

1. Consider the counter-clockwise torque due to the string tension as positive, and the clockwise torques due to the weight of the arm and the load as negative.

8. Calculate the percent difference from the ratio of the net torque to the counterclockwise torque.

### Setup 2

1. Reposition the metal rod at an angle. θ should be about 50°. Make sure the wooden arm is still parallel to the ground. (Figures 4 and 5)

2. Measure and record the new angle θ.

3. In Capstone click Record and record the tension of the string, T.

4. Compute the net torque about the elbow joint, just as you did in Setup 1.

5. Calculate the percent difference from the ratio of the net torque to the counter-clockwise torque.

Concept Checkpoint 2:

1. Discuss with your partner which of the two arm positions, Setup 1 or Setup 2, is more difficult for you to hold a weight. In which case is the “applied tension” your muscle gives the greatest?

2. In which case is the “applied tension” more nearly perpendicular to the length of your arm?

3. Does this make sense? Why?

4. Call over a TA or instructor and explain your conclusion.

### Part B: Estimating the Tension in a Muscle

1. Knowing the results from setups 1 and 2, determine how you should position your arm if you want to hold a weight most easily.

1. hint: which setup has the least tension in the force sensor?

2. Try this yourself with a 1 kg mass and make a note in your e-journal.

3. Using the directions for Setup 1, estimate the muscle tension needed to raise a weight of about 10 N. Use reasonable estimates of angle, weight, and length to calculate this.

1. The diagram above has estimates you can use.

### Part C: Catwalk (Bridge) and Cantilever

 Catwalk (Bridge) Setup
1. Plug a second force sensor into channel B. In Capstone indicate a force sensor in channel B.

2. Remove the I-hooks from both force sensors. Set them about 60 cm apart from each other, with the metal plates upwards. (Figure 9).

3. Calibrate both Force Sensors.

1. NOTE: For your second calibration point, set a 1.96 N (200 grams) weight directly on the metal plate of each sensor (you will have a 200 g weight on both sensors at the same time) and read from sensor after inputting the 1.96 N.

4. Place two Digit Display's on the Main Screen.

5. Measure and record the mass of a meter stick.

6. Lay the meter stick horizontally across both force sensors, positioned with one sensor at the 10 cm mark and the other at the 70 cm mark.

1. Make sure that the meter stick is horizontal.

7. Place a 500 g mass (4.9 N) load at the 60 cm mark.

8. Record the force reading for each force sensor.

#### Analysis:

The vector sum of the forces should be zero (within experimental error) for static equilibrium.

1. Compute the vector sum of all vertical forces.

2. Compute the percent difference by comparing the nearly zero net force with the upward force.

3. Sum the four torques about the 0 cm point, including the counter-clockwise torques due to both upward forces F1 and F2 and the clockwise torques due to the weights of both the stick and the 4.9 N load.

4. Calculate the percent difference.

### Setup 3

(2 Extra Credit Points Possible)

1. Reposition the metal rod vertically (perpendicular to the floor) and the wooden arm at a slight upwards angle, about 40° above the horizontal. Angle θ should be greater than 90°. (Figures 6 and 7)

2. Use the protractor to measure and record the angles θ and a.

3. In Capstone click Record and record the tension of the string, T.

4. Compute the net torque.

### Part D: Cantilever Procedure

(2 Extra Credit Points Possible)

 Cantilever Setup
1. Invert the force sensor at the 10 cm mark.

2. Recalibrate the inverted force sensor. You will need to temporarily reattach the hook to hang the 500 gram weight (4.9 N) for the calibration.

3. Take the hook off and adjust the level of the inverted force sensor so that the meter stick is level.

4. Carefully center the inverted force sensor at the 10 cm mark. Check that the second force sensor is still upward at the 70 cm mark in the figure of the Cantilever Setup above.

5. Place the 4.9 N load at the 90 cm mark.

6. Record the force reading for each force sensor.

#### Analysis:

1. Sum the vertical forces.

2. Compute the percent difference from the ratio of the net force to the total upward force.

3. Sum the torques about the 0 cm point including the single (positive ) counter-clockwise torque due to the upward force F2 and the three (negative) clockwise torques due to F1, the weight of the stick and the 4.9 N load.

4. Calculate the percent difference.

## Layout: /2

• Title:

• Names: (Put 'scribe' beside the name of the person writing the abstract, conclusion, etc. This will be a different person each week as you and your lab partner exchange duties!!)

• Date

• Time In & Out:

## Preliminaries: /4

• Personalized Statement of Objectives:

• Methods Used: (Insert a labeled webcam image of apparatus. Describe what and how measurements are made.)

• Predictions Made: (Include sketches, plots, and descriptions of expected results.)

## Data: /8 and Results: /6

### Setup 1

1. Describe your data collection techniques for investigating the torque on the horizontal forearm(include a picture of your setup).

2. Record the mass and weight of the wooden arm

1. marm =

2. Warm = marm*g =

3. Record the distance from the elbow joint to the point where the force sensor is tied to the arm:

1. dtension =

4. Record the distance from the elbow joint to the center of gravity of the forearm:

1. darm =

5. Record the distance from the elbow joint to the point of application of the load:

6. Record the mass and weight of the load:

7. Record angle θ.

8. Record the tension from Capstone.

9. Compute and record the net torque about the elbow joint

1. Consider the counter-clockwise torque due to the string tension as positive, and the clockwise torques due to the weight of the arm and the load as negative.
$$\tau_ {Net} = |\tau_ {ccw}| - |\tau_ {cw}| = Td_{T} sin(\theta) - W_{arm}d_{arm} - W_{load}d_{load}$$

10. Calculate and record the percent difference from the ratio of the net torque to the counterclockwise torque.
$$\%Diff = \frac{|\tau_{Net}|}{|\tau_{ccw}|} \times 100\%$$

11. In an ideal world with perfect measuring devices, what value would you expect for the net torque?

### Setup 2

1. Describe your data collection techniques for investigating the horizontal forearm with vertical upper arm (include a picture of your setup).

2. Record angle θ.

3. Record the tension from Capstone.
T=

4. Compute and record the net torque about the elbow joint.
$$\tau_ {Net} = |\tau_ {ccw}| - |\tau_ {cw}| = Td_{T} sin(\theta) - W_{arm}d_{arm} - W_{load}d_{load}$$

5. Calculate and record the percent difference from the ratio of the net torque to the counter-clockwise torque.
$$\%Diff = \frac{|\tau_{Net}|}{|\tau_{ccw}|} \times 100\%$$

### Part B: Estimating the Tension in a Muscle

1. Of the two arm setups, which involves the least amount of muscle tension? Why?

1. hint: which setup has the least tension in the force sensor

2. Try holding weights at different angles and positions, write your observations:

3. Estimate the muscle tension in your biceps using setup 1.

1. Record all of your estimations:

1. Tbisep =

### Part C: Catwalk (Bridge) and Cantilever

1. Describe your approach to determining forces and torques on a horizontal meter stick (include an image).

2. Measure and record the mass of a meter stick.

1. mstick =

2. Wstick = mstick*g =

3. Measure and record the mass, weight, and position of the load.

4. Record the force reading for each force sensor.

1. F1 =

2. d1 =

3. F2 =

4. d2 =

5. Compute and record the vector sum of all vertical forces using:
$$F_{net} = \sum F_{y} = F_{1} + F_{2} - W_{stick} - W_{load}$$

6. Compute and record the percent difference by comparing the nearly zero net force with the upward force:
$$\%Diff = \frac{|F_{net}|}{F_{1} + F_{2}} \times 100 \%$$

7. Sum and record the four torques about the 0 cm point. Include the counter-clockwise torques due to both upward forces F1 and F2 and the clockwise torques due to the weights of both the stick and the 4.9 N load:
use:
$$\tau_{net} = |\tau_{ccw}| - |\tau_{cw}| = F_{1}d_{1} + F_{2}d_{2} - W_{stick}d_{stick} - W_{load}d_{load}$$

8. Calculate and record the percent difference from:
$$\%Diff = \frac{|\tau_{net}|}{|\tau_{ccw}|} \times 100 \%$$

1. Describe your data collection techniques for investigating the acutely angled forearm with vertical upper arm (include a picture of your setup).

2. Record angle a. (the angle from the horizontal to the wooden arm)

3. Record angle θ.

4. Record the tension from Capstone
T=

5. Compute and record the net torque using:
$$\tau_ {Net} = Td_{T} sin(\theta) - W_{arm}d_{arm}sin(b) - W_{load}d_{load}sin(b)$$

1. Describe your approach to determining forces and torques on a horizontal meter stick (include an image).

2. Measure and record the mass, weight, and position of the load.

3. Record the force reading for each force sensor.

1. F1 inverted =

2. d1 inverted =

3. F2 =

4. d2 =

4. Compute and record the vector sum of all vertical forces using:
$$F_{net} = \sum F_{y} = F_{2} - F_{1} - W_{stick} - W_{load}$$

5. Compute and record the percent difference from the ratio of the net force to the total upward force using:
$$\%Diff = \frac{|F_{net}|}{|F_{2}|} \times 100 \%$$

6. Sum and record the torques about the 0 cm point. Include the single (positive ) counter-clockwise torque due to the upward force F2 and the three (negative) clockwise torques due to F1, the weight of the stick and the 4.9 N load.
use:
$$\tau_{net} = |\tau_{ccw}| - |\tau_{cw}| = F_{2}d_{2} + F_{1}d_{1} - W_{stick}d_{stick} - W_{load}d_{load}$$

7. Calculate and record the percent difference from:
$$\%Diff = \frac{|\tau_{net}|}{|\tau_{ccw}|} \times 100 \%$$

## Conclusion: /4

1. Percent differences… why?

2. What would be different in a perfect system?

3. Is Net torque 0 for a moving arm?

4. etc…

## Abstract: /4

This is a formal statement of what this laboratory experiment was all about.

Included in this paragraph should be something about:

• The Objectives

## Certification: /2

• Include a statement that the work done in this lab and submitted in this report is yours and your partners.

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