# The Skater

 Materials: Computer and School Network Time Allotment: 3 Class Days

### Introduction:

You have seen that any object moving in a circle has a changing velocity. The object moves with a constant speed, but the direction of the velocity changes; this velocity change corresponds to an acceleration which is directed towards the center of the circle. This inwards (or centripetal) acceleration is caused by an unbalanced force which is directed towards the center of the circle (and thus, commonly referred to as a centripetal force). In this activity you will simulate the motion of a skater skating in circles on a rink measuring 200 m by 200 m. The simulation will portray the motion as seen from above.

### Getting Ready:

1. Log on to the school server in the usual manner.
2. Open the Physics Explorer application (One Body model) by double-clicking on its icon; it is found in the directory Science-Math/Science Apps/Physics.
3. Choose Open... from the File menu and navigate to the Unit 6 folder.
4. Open the file titled Skater. Check that the Velocity Window values appear as below.

### Speed and Radius:

According to Newton's first law, it is the natural tendency of objects to maintain straight-line motions (i.e., an inertial path). A skater deviates from this straight-line path and moves in a circle by leaning on the ice at an angle to the vertical. This lean causes a reaction force by the ice which is directed inward towards the center of the circle. Since the acceleration of a skater is related to the centripetal force acting on the skater, one way to measure acceleration is by measuring the angle she needs to lean toward the center of the circle in order to keep from moving in a straight line (tangent to the circle). This portion of the lab has a pre-set, fixed acceleration. This is reflected in the constant "angle of lean" shown in the output display.
1. Vary the skater's starting speed ("Velocity Magnitude") and run the simulation to find the largest circle and the fastest speed she can move with while still maintaining the circular path around the ice rink (without striking the walls of the rink).
Fastest speed for largest circle: _____________ m/s

Radius of largest circle: _____________ m

2. The skater decides to try skating in circles at different speeds over the range of 3 m/s to 11 m/s, with the acceleration constant (due to the constant angle of lean).
• Run the simulation for these various speeds.
• Use the RULER tool (see diagram) to help measure the radius of the circles. (Use the RULER tool to mark off a diameter of the circle. The radius measured will appear in the Message window.)
• To record speed and radius to the spreadsheet, be sure to start with all cells clear. After making each measurement of radius, click on the "Record Speed and Radius Data" button, which transfers to the spreadsheet the speed and the last measurement you made with the RULER tool. (NOTE: Click the button only when you wish the measurement to appear in the graph.)
• Also record your values in the table below.

### Radius (m)

11

________________

8

________________

6

________________

4

________________

3

________________

3. Open the spreadsheet from the Windows menu. Generate a graph of your results by clicking on the "Plot the Graph" button. (If necessary, highlight spreadsheet columns B-D, choose the "Line Graph" option from the Plot menu; check the "Graph All Data" boxes for both the x and the y-axis, and click "OK.") The graph displays the radius vs. speed. Describe the shape of the graph (linear or curved; upward sloping or downward sloping; if curved, does the slope increase or decrease with increasing speed; etc.).

4. Try to find a function of the radius which yields a straight line when plotted against speed. That is, determine to which power that the radius must be raised in order to achieve a linear relation for your data set. To do this, follow this procedure:
• Calculate the values for your function in the spreadsheet. Do this by adding a new test formula to column E, using the "Spreadsheet Sampling" option in the Edit menu.
• Highlight spreadsheet columns B-E.
• Click on the "Plot the Graph" button. (If necessary, highlight spreadsheet columns B-E, choose the "Line Graph" option from the Plot menu ; check the "Graph All Data" boxes for both the x and the y-axis, and click "OK.")

When you plot the graph, your function is superimposed on the original curve of speed vs. radius.
What is the relation between radius and speed when an object is moving in a circle with a constant acceleration? State both the qualitative relation and the mathematical relation (i.e., the equation).

### Acceleration and Radius:

1. To examine how the radius of the skater's circle and the acceleration of the skater are related, click on the "Acceleration Window" button. This window allows you to hold velocity constant while varying acceleration and radius. While the skater herself might not be conscious of the values of her acceleration, the angle of lean is certainly a variable which she can control. You will measure the angle of lean (which is indicative of the acceleration) in this portion of the lab and relate it (and the acceleration) to the radius of the circle which the skater moves in.

2. Clear all the cells of the spreadsheet! Click on the "Trace On" button to turn on the trace of the path. Vary the acceleration of the skater through the range from 0.8 m/s/s to 4.0 m/s/s. Keep the velocity constant at 8.0 m/s. Find the values of the radius in each case.
• After each "run" of the simulation, fill in the table below.
• Also click on the "Record Acceleration and Radius Data" button to record the acceleration and radius data to the spreadsheet.

### Radius (m)

_____________

_____________

_____________

_____________

_____________

_____________

_____________

_____________

_____________

_____________

3. Now select the two columns of data and click on the "Plot the Graph" button to generate a graph of the data you have just gathered. (If necessary, highlight spreadsheet columns B-D, choose the "Line Graph" option from the Plot menu; check the "Graph All Data" boxes for both the x and the y-axis, and click "OK.")

What is the relationship between the acceleration and the radius for a constant starting speed?

4. In practice, how does a skater move in a tighter circle? Does she lean with more or less angle? Does she slow down or speed up? Does she push a button on her skates? In the space below, explain what a skater consciously does to skate in a circle with a smaller radius.

### Relation between Acceleration, Speed and Radius:

From the relations you found in the previous two sections, propose a relationship between acceleration, speed, and radius for an object moving in a circle.

### Force and Centripetal Acceleration:

You know from Newton's second law that Fnet=ma, so that if the skater is accelerating, there must be a force acting on her in the direction of her acceleration. Physics Explorer shows you the acceleration vector as the skater moves in a circle.

1. In what direction is the force acting? ____________________

2. If the skater has a mass of 50 kg, what force acted when she moved in her first circle (a = 0.8 m/s/s)? PSYW

3. From your answer to the section on acceleration, speed, and radius, what equation relates the force acting on a mass m, moving in a circle with constant speed v and radius r?

### Extensions:

The goal of this unit is to combine an understanding of Newton's second law, circular motion concepts, and vector concepts to analyze complesx situations involving the motion of objects in circles. Express this understanding by answering the following concepts.

1. In the space at the right, draw a free-body diagram on the skater as she moves in a circle at constant speed. [Hint: The contact force between the skater and the ground has two components: one perpendicular to the ice (normal) and the other directed to the center of the circle (friction). These two vectors add together to produce a vector directed in the direction of lean.]

2. If a skater has a mass of 75 kg, a velocity of 10 m/s and moves in a circle with a radius of 45 m, then determine the ...

1. ...acceleration.

2. ...net force.

3. ...weight of the skater.

4. ...normal force.

5. ...friction force.

6. ...the angle of lean.

3. A 0.05-kg toy airplane is suspended from the ceiling by a string. The string makes an angle with the vertical due to the fact that it pulls upwards to balance the weight of the plane and inward to provide the centripetal force to sustain the circular motion. In the space below, construct a free-body diagram showing the forces acting upon the toy airplane.

4. The toy airplane sweeps out a 1-meter radius circle in 1.4 seconds. Determine the ...

1. ...velocity.

2. ...acceleration.

3. ...net force.

4. ...weight of the plane.

5. ...vertical component of tension.

6. ...horizontal component of tension.

7. ...angle which the plane makes with the horizontal.

### Conclusion:

Write a succinct and organized paragraph, discussing the major concepts learned in this lab. Do a bang-up job!

This page created by Tom Henderson and last updated on 8/7/97.