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



Physics 211 Lab

The Kinematics and Dynamics of

Circular and Rotational Motion

Introduction:

When discussing motion, it is important to be aware that there are several different types and that each type is dependent upon a particular frame of reference. Linear motion is the most basic form, involving the translation of a particle from one point to another in one dimension. Projectile motion, although two-dimensional, is easily analyzed by separating it into its linear components. Circular motion is a special case of two-dimensional motion in which an object translates in a circular path. Rotational motion is observed when an object itself rotates about some internal axis. Consider the following examples:

• A car traveling along a straight road exhibits linear motion from the perspective of a person standing on the side of the road. The wheels of the car exhibit rotational motion about the axle. A pebble embedded in the tread of the car exhibits circular motion about the axle.

• A baseball spins towards a batter at home plate. From the perspective of a spectator, the ball is undergoing translational motion across the infield. However, the ball itself is rotating about its own axis. The stitches of thread on the outside of the ball are moving in a circle relative to the spinning axis of the ball. (These circles differ depending upon the location of the particular stitch.)

• The earth rotates on its own axis as it also moves in a circle from the perspective of a stationary sun. In addition, people located at the surface of the earth undergo a different circular motion each day around the axis of rotation of the earth. (These circles differ depending upon the latitude of each person.)

It is important to notice that circular motion connects the concepts of linear and rotational motion. For any object that is rotating, a particular point on that object is moving in a circle. One of the goals of this lab activity is to explore and understand this connection.

The translational motion of any point particle can be described in terms of standard Cartesian coordinates. In other words, Cartesian coordinates can describe both linear and circular motion. However, in the case of circular motion, the particle moves an arc length, s, around a circle with a (constant) radius, r. Therefore, in this case it is simpler to use Polar coordinates where the position of the particle can be specified by r and the angular position, θ, rather than x and y.

Notice that the right triangle defines the relationship between the location of the particle in Cartesian and Polar coordinates. The following equations describe how to transform between Polar and Cartesian coordinates.

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The radius of a particle undergoing circular motion is always a constant. The angular position, however, will change with time depending on the motion of the particle. Since only the angular position changes with time its behavior is exactly analogous to the behavior of the position in one-dimensional motion that was studied previously. Thus, the angular equivalent of the kinematic quantities for one-dimensional motion can be defined as follows:

Arc distance traveled [pic] Angular position

Linear (tangential) velocity [pic] Angular velocity

Linear (tangential) acceleration [pic] Angular acceleration

The relationships between the angular position, velocity, and acceleration are exactly the same as the relationships previously determined for one-dimensional motion. For example, for motion with constant angular velocity, ω:

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For motion with constant angular acceleration, α :

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The Net Force that causes an object of mass, m, to move in a circular path is called the centripetal force, Fc. At a particular constant linear speed, v, the following equation (Newton’s Second Law) describes the dynamics of the object’s circular motion:

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where the magnitude of the centripetal acceleration is given by:

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For circular motion that is not constant, the total acceleration (magnitude and direction) of the object, at every moment, is determined by adding the centripetal (radial component) and linear (tangential component) accelerations together. Note: ac and at are vectors that form a right triangle when added together because they are perpendicular to each other.

[pic]

Remember that tangential acceleration is a consequence of any change in the linear (and therefore, angular) speed of the object. Centripetal acceleration is a consequence of the rate at which the direction of the object changes at every moment.

The centripetal force, the component of the net force directed towards the center of the circle, is caused by different types of forces. Here are some examples for the case of a single force causing circular motion:

• Tension in a string (such as when a ball is whirled in a horizontal circle at the end of a rope)

• The normal force (such as on the Rotor ride found at many amusement parks)

• Gravity (such as the orbits of planets around the sun)

• Static friction (such as a car traveling around a level curve)

Here are some examples for the case of two forces in combination causing circular motion:

• Tension and gravity (such as when a ball is whirled in a vertical circle at the end of a rope)

• The normal force and gravity (such as when a person rides a vertical loop on a roller coaster)

• The normal force and static friction (such as a car traveling around a banked curve)

In this lab, you will explore the case of a penny moving in a circle on a level surface due to the force of static friction. Recall that static friction is a variable force, able to provide resistance up to a particular maximum value. At this point, the object is said to be on the verge of slipping. The following equations describe the relationship between static friction and the dynamics of the non-constant circular motion at this moment.

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where, on a level surface, the normal force is equal in magnitude to the force of gravity. Remember that the magnitude of atotal is determined by taking a vector sum.

Physics 211: Lab

The Kinematics and Dynamics of

Circular and Rotational Motion

Goals:

• Compare the graphs of circular motion for a rotating turntable.

• Determine the coefficient of static friction between the turntable surface and a penny.

• Predict the magnitude of the angular velocity that causes a penny at a given radius to slip.

Equipment List:

Rotating Platform with attached Rotary Motion Sensors

Stickers located at two different radii

Pulley

String

Hanging mass and hanger

Penny

Ruler

Computer & Equipment Set Up:

1. Start by making certain that the string used to turn the turntable is not attached to any hanging mass.

2. Set up Science Workshop( to read the data collected from the Rotary Motion Sensor. You will need to change the default settings of the sensor; Double click on the rotary motion sensor icon in the experiment setup window – this opens up the sensor properties window. In the sensor properties window select the measurement tab and un-check the box marked angular position (deg) and check the box that reads angular position (rad), angular velocity (rad), and angular acceleration (rad).

3. Create a graphing window to display Angular Position (θ) vs. Time.

4. Check the calibration of the sensor: Press Record; rotate the turntable exactly once; Press Stop; look at the Angular Displacement values recorded on your graph. (They are measured in “radians”.) Decide whether or not the graph verifies that your turntable is correctly calibrated. (If it is not, see your TA immediately.) Import your graph to the Word template and clearly explain your decision.

Activity 1: Kinematics of Circular Motion

The purpose of Activity 1 is to compare the graphs of rotational motion for an object, the turntable, which starts from rest and rotates with constant angular acceleration.

1. Add to the Angular Velocity and Angular Acceleration to the Angular Position (θ) vs. Time graphing window. You should now have all three graphs in one window.

2. Carefully wind the string around the base of the turntable. Place the string over the pulley and attach the hanging mass (use 100 grams or 150 grams) to the other end of it as shown in the picture above.

3. Press Record and, releasing the turntable from rest, gather data describing the rotational motion of the turntable as the mass falls. The angular acceleration of the turntable should be relatively constant.

4. Using the “Statistics” capabilities of Science Workshop(, calculate the Angular Acceleration, α, of the turntable using two different methods. Explain each of your methods and state your results.

5. Copy the graphing window (including the statistics information that you calculated) into the Word template by using “Paste Special”. Paste each as if it were a “picture”.

6. Compare your three graphs and explain, mathematically, how…

• …the Angular Position vs. Time graph & the Angular Velocity vs. Time graph are related to each other.

• …the Angular Velocity vs. Time graph & the Angular Acceleration vs. Time graph are related to each other.

Activity 2: The Dynamics of Circular Motion

There are two purposes for Activity 2. The first is to use the results of a penny slipping off of the turntable at a particular radius to determine the coefficient of static friction between the surface of the turntable and a penny. The second is to use this value of the coefficient to predict the angular velocity at which the penny will slip when placed at a different radius.

Part I: Determine the Coefficient of Static Friction on the Turntable

1. Measure the radius of the circle created by the outer blue sticker, Ro. Record this value in the table below.

2. Carefully rewind the string around the base of the turntable and place the string over the pulley with the hanging mass attached. (Note: The size of the hanging mass must be large enough to cause a penny to slip at both the blue and yellow positions, but small enough to keep it from slipping too quickly. Recommended: 150 grams)

3. Place a penny at a distance, Ro, from the center of the turntable. [Note: Do not place the penny directly on top of the blue sticker.]

4. Press Record and, releasing the turntable from rest, gather data describing the rotational motion of the turntable as the mass falls. Using your hand, stop the turntable at the very moment the penny slips from the surface. Then, Press Stop to end the collection of data.

5. From reading your graph of Angular Velocity vs. Time, determine the magnitude of the Angular Velocity of the turntable at the moment just prior to when the penny slipped. Record this value in the table below.

6. Record the value of the Angular Acceleration, α, of the turntable (determined in Activity 1) in the table below. (Note: This value should agree with your current data.)

7. For the moment just prior to when the penny slipped, calculate the linear (tangential) velocity, the tangential acceleration, and the centripetal acceleration of the penny. (Note: The tangential acceleration will likely be much smaller in magnitude than the centripetal acceleration.) Record your results and explain your calculations in the table below.

8. Determine the coefficient of static friction between the turntable surface and the penny. Record your results and explain your calculations in the table below. Clearly and completely explain your method of calculating μs.

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

|Ro = Radius (meters) | |This radius was measured using a ruler. |

|ω = Angular Velocity (radians/s) | | |

|α = Angular Acceleration (radians/s2) | | |

|V = Linear Velocity (m/s) | | |

|at = Tangential Acceleration (m/s2) | | |

|ac = Centripetal Acceleration (m/s2) | | |

|atotal = Total Acceleration (m/s2) | | |

|μs = Coefficient of Static Friction | | |

Part II: Predict the Angular Velocity at which the Penny will Slip at a Different Radius

9. Measure the radius of the circle created by the inner yellow sticker, Ri. Record this value in the table below.

10. Record the value of the coefficient of static friction (calculated in Part I) in the table below.

11. Predict the Angular Velocity of the turntable that will cause the penny to slip when placed at a distance, Ri, from the center of the turntable. Clearly and completely explain your method of calculating ω.

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

|Ri = Radius (meters) | |This radius was measured using a ruler. |

|μs = Coefficient of Static Friction | |See Table in Part I. |

|ω = Predicted Angular Velocity | | |

|(rad/s) | | |

12. Test your prediction: Place a penny at a distance Ri from the center of the turntable. [Note: Do not place the penny directly on top of the yellow sticker.] Press Record and, releasing the turntable from rest, gather data describing the rotational motion of the turntable as the mass falls. Using your hand, stop the turntable at the very moment the penny slips from its position. Then, Press Stop to end the collection of data.

13. From reading your graph of Angular Velocity vs. Time, determine the magnitude of the Actual Angular Velocity of the turntable at the moment just prior to when the penny slipped. Record this value in the table below.

|ω = Actual Angular Velocity | |

|(rad/s) | |

14. By what % does your Predicted value differ from the Actual value? Show your calculation in addition to your final answer. Does the % difference seem reasonable? Can you account for this difference in terms of the inaccuracy of your measurements? Explain.

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θ

r

Y

X

(x, y) = (r cosθ , r sinθ)

S

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