Lab #8: The Kinematics & Dynamics of Circular & Rotational ...



Lab #10: Template

Rotational Motion in 3-Dimensions

Score: _____ / 75

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Student Name:

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Computer & Equipment Set Up: (18 pts)

The purpose of this part of the lab is to become familiar with the gyroscope and learn how to correctly interpret the data taken by the Rotary Motion Sensor.

4. Note the motion of the gyroscope’s disk as it continues to spin freely and answer the following qualitative questions:

• How does the magnitude of the angular velocity of the disk change over a relatively large time interval? (2 pts)

• How does the magnitude of the angular velocity of the disk change over a relatively small time interval? (2 pts)

• What does the motion of the disk tell you about the amount of torque exerted on the disk due to kinetic friction in the axle? (2 pts)

5. Data is collected by placing the Rotary Motion Sensor on the outer rim of the gyroscope’s disk. Practice taking this data and note the shape of the graph. Determine which data points represent the motion of the sensor when it is in rolling along the rim of the gyroscope’s disk. Answer the following questions:

• Why does the angular velocity of the sensor decrease over time as it is held against the rim of the gyroscope collecting data? (2 pts)

• What is happening to the angular velocity of the gyroscope’s disk over this same time interval of data collection? Why? (2 pts)

• What techniques should you employ when taking this data so that the motion of the gyroscope’s disk remains as unaltered by the collection of this data as possible? Which y-value will you choose to represent the Angular Velocity of the Rotary Motion Sensor? Explain. (2 pts)

6. (2 pts) Measure and record the radius of the gyroscope’s disk and record it in the table below. The radius of the Rotary Motion Sensor’s disk has already been carefully measured for you using a caliper.

|Radius of Gyroscope’s Disk |Radius of Rotary Motion Sensor’s disk |

| |2.759 cm |

7. (4 pts) Write an equation to determine the Angular Velocity of the Gyroscope’s Disk given the Angular Velocity of the Rotary Motion Sensory. (Hint: When in contact, the points on the edge of the disk and the sensor have the same linear speed, v.) Ask your TA for help if you have trouble deriving this equation, as it is necessary for Activity 2.

|ωof disk = |

Activity 1: Qualitative Observations of the Gyroscope (22 pts)

The purpose of this activity is to observe that the Right Hand Rules of 3-D Rotation correctly predict the motion of the gyroscope.

1. Position your gyroscope on the table so that it is oriented in the same direction as the picture below.

[pic]

2. (2 pts) The X and Y axes are defined above. In which direction does the +Z axis point? All answers in this lab should be consistent with these coordinate axes. Notice that the pivot point is located at the origin in the above photo. (If you are unsure of the Right Hand Rule needed to answer this question, then ask your TA.)

3. (20 pts – 2 pts each) From the viewpoint shown in the picture above, complete the following table by clearly writing your observations and an explanation. Use reference to the X,Y, &Z axes and the appropriate Right Hand Rule.

|Tasks |Questions |Observations / Explanations |

|(1-a) While the apparatus is at rest, apply|Relative to the pivot, what is the | |

|a downward (-Y direction) force to the left|direction of the Torque created by the | |

|end of the rod. |force? | |

|(1-b) While the apparatus is at rest, apply|Relative to the pivot, what is the | |

|an upward (+Y direction) force to the left |direction of the Torque created by the | |

|end of the rod. |force? | |

|(1-c) While the apparatus is at rest, apply|Relative to the pivot, what is the | |

|a downward (-Y direction) force to the |direction of the Torque created by the | |

|right end of the rod. |force? | |

|(1-d) While the apparatus is at rest, apply|Relative to the pivot, what is the | |

|an upward (+Y direction) force to the right|direction of the Torque created by the | |

|end of the rod. |force? | |

|(2-a) Rotate the disk so that the top edge |What is the direction of the angular | |

|of the disk is moving in the +Z direction. |momentum of the disk? What is the direction| |

| |of the angular velocity of the disk? | |

|(2-b) ) Rotate the disk so that the top |What is the direction of the angular | |

|edge of the disk is moving in the -Z |momentum of the disk? What is the direction| |

|direction. |of the angular velocity of the disk? | |

|(3-a) While the disk is rotating as in |Describe what happens to the motion of the | |

|(2-a), hang a mass from the left end of the|gyroscope relative to the pivot. | |

|rod. | | |

|(3-b) While the disk is rotating as in |Describe what happens to the motion of the | |

|(2-b), hang a mass from the left end of the|gyroscope relative to the pivot. | |

|rod. | | |

|(4-a) Repeat the set up in (3-a): spin the |How does the angular speed of the disk | |

|disk quickly and then spin it slowly. |affect the motion of the gyroscope around | |

| |the pivot as observed in (3-a)? | |

|(4-b) Repeat the set up in (3-b): spin the |How does the angular speed of the disk | |

|disk quickly and then spin it slowly. |affect the motion of the gyroscope around | |

| |the pivot as observed in (3-b)? | |

Activity 2: Quantitative Observations of the Gyroscope – Newton’s Second Law (20 pts)

The purpose of this Activity is to verify Newton’s Second Law (in Angular Form). The Introduction of this lab provides a complete explanation of the theory used to collect the data below.

(15 pts)

|Quantity |Result |Explanation of how Result was obtained… |

| |(magnitude) | |

|M = total mass hanging from lever arm (kg) | | |

|F = weight of hanging mass (N) | | |

|r = lever arm (m) | | |

|(distance from pivot to hanging mass) | | |

|θ = Angle between vectors F and r (() | | |

|( = Torque (N m) | | |

|(caused by hanging mass relative to pivot) | | |

|I = Rotational Inertia of Disk (kg m2) |.0136 |This value was measured for you. |

|ω = Angular Velocity of Disk (rad/s) | | |

|L= Angular Momentum of Disk (kg m2/s) | | |

|Δθ = Angular Displacement of Rod (rad) |2π |This value was chosen for you. If you would like to use |

|(about the Y axis) |(360 () |another value, simply make note of it. |

|Δt = time it takes the rod to rotate Δθ (sec) | |Use the Record and Stop buttons on Science Workshop( to |

| | |measure the time it takes the gyroscope to rotate 360 ( |

| | |around the Y axis. |

|ωave = Angular Velocity of Rod (rad/s) | | |

|(Caused by the direction of ω changing) | | |

|L ωave : which equals ΔL/Δt (N m) | | |

|(Derivation was shown in the Introduction.) | | |

1. (5 pts)Decide whether or not the results above support Newton’s Second Law (in Angular Form). Explain your reasoning clearly and completely. Defend your position by calculating a percent difference and interpreting the meaning of it.

Activity 3: Conservation of Angular Momentum (15 pts)

1. (2 pts) Using the appropriate Right Hand Rule, complete the table below.

|Initial Angular Momentum (kg m2/s) of … |direction |

|The Bicycle Wheel | |

|The Person (and stool) | |

2. (2 pts) Quickly turn the wheel upside down so that its axis of rotation is again vertical. Record the resulting motion in the table below.

|Final Angular Momentum (kg m2/s) of … |direction |

|The Bicycle Wheel | |

|The Person (and stool) | |

3. (11 pts) Assume that (1) Angular Momentum is conserved and (2) the value of the Angular Momentum of the Bicycle Wheel given below is correct. Complete the rest of the tables below, using the directions observed above. Hint: A vector addition diagram is extremely helpful.

|Initial Angular Momentum (kg m2/s) of … |magnitude |direction |

|The Bicycle Wheel |.294 | |

|The Person (and stool) | | |

|The Total Momentum | | |

|Final Angular Momentum (kg m2/s) of … |magnitude |direction |

|The Bicycle Wheel | | |

|The Person (and stool) | | |

|The Total Momentum | | |

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+X

+Y

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