Curve Ahead, Reduce Speed!

Introduction

Whether riding a bicycle or driving a car, people quickly learn that rounding a curve too fast can be dangerous! Demonstrate the maximum speed an object can safely travel along a curved path by analyzing the effects of variables such as the radius of the curve, banking the curve, and friction on the rotational motion of a plastic block. This simple and practical demonstration will develop students’ understanding of centripetal force, speed and acceleration and help them connect these concepts to everyday events.

Concepts

• Centripetal force
• Linear speed versus rotational speed
• Newton’s laws of motion

Materials

Safety Precautions

Use caution when performing this activity. Practice the demonstration before performing it in the classroom. Both the demonstrator and all observers should wear protective eyewear. Have all observers remain a safe distance away as the demonstrator rotates the turntable and PVC block—the block may fly off the turntable. Follow all laboratory safety guidelines.

Prelab Preparation

1. Cut the sandpaper strip into one 3" x 6" piece and two 3" x 3" pieces.
2. Peel off the backing from one of the 3" x 3" pieces of sandpaper and press the adhesive side down on the slope of the wood ramp.
3. Peel off the backing from the 3" x 6" piece of sandpaper. Press the adhesive side down along the radius of the turntable, with one end of the sandpaper as close to the outer edge of the turntable as possible (see Figure 1).
4. Peel off the backing from the second 3" x 3" piece, fixing this piece onto the turntable along the same diameter as the longer piece, on the opposite side of the turntable (see Figure 1).
5. Determine the center of the turntable by measuring the diameter of the circle and lightly marking the halfway point with a pencil line. Measure again, using a diameter perpendicular to the first diameter measured, again lightly marking the halfway point with a pencil line. The center should be where the two lines intersect. Note: The intersection of the pencil marks will be on the longer sandpaper strip, near the inner edge (see Figure 1).

Procedure

Part I. Control

1. Place the PVC block lengthwise along the edge of the turntable, on top of the longer sandpaper strip (see Figure 2). Explain to the students that the block on the turntable represents a car going around a curve, and the sandpaper under the block is providing traction, or friction, just as contact between the car tires and the road provides friction that helps prevent a car from sliding off the road.
2. Find the radius of the circular path the “car” will be traveling by measuring from the center of the turntable to the center of the block (see Figure 2).
3. Instruct students to record the radius in the data table for Part I on the worksheet.
4. Using the shorter piece of sandpaper as a touch point, begin to slowly rotate the turntable.
5. Gradually increase the speed of rotation by lightly touching the sandpaper each time it comes around and pushing in the direction the turntable is rotating. Note: A light touch is all that is needed. Pushing downward with too much force will cause the turntable to wobble.
6. Continue to increase the rotational speed of the turntable until the block begins to slide off the turntable. Caution: Increasing the speed too quickly may cause the block to fly off the turntable. Observers should be a safe distance away!
7. Replace the block on the turntable at the same location as in step 1 and repeat steps 4–5, reaching the maximum speed possible with the block remaining in place. This is the maximum “safe” speed of the block on the turntable.
8. Once a constant maximum speed is achieved, use a timer to measure the time it takes for 10 revolutions of the turntable. Note: Count the revolutions out loud each time you touch the sandpaper. Start the timer on the count of zero and stop the timer on the count of ten (see the Tips section for ways to involve students and ensure accurate timing).
9. Instruct students to record the time for 10 revolutions in the data table for Part I on the worksheet.
10. Repeat steps 7–9 two more times.

Part II. Tighter Curve

11. Place the block on the long sandpaper strip so the center of the block is closer to the center of the turntable than in Part I (a radius of 10 cm or less). Explain to the students that the “car” will now be rounding a “tighter turn.”
12. Have students predict what effect the decreased radius of the curve will have on the maximum safe speed of the block as it rotates with the turntable compared to Part I.
13. Follow the same procedure as in steps 2–10 of Part I. Students should record the data in the table for Part II. The radius will be smaller than it was in Part I and the time for 10 revolutions should also be less. This may surprise the students. As they later work through the calculations, they will discover the difference between rotational speed (revolutions/time) and linear speed (distance/time).

Part III. Banked Curve

14. Remove the block from the turntable and place the wood ramp—sandpaper side up—on top of the longer sandpaper strip at the edge of the turntable (see Figure 3)
15. Use masking tape to secure the front and back edges of the ramp to the turntable.
16. Place the block lengthwise across the top of the ramp. The distance from the center of the circle to the point directly below the center of the block as it sits on the ramp should be the same as the radius of the circle in Part I.
17. Have students predict what effect the “banked curve” will have on the maximum safe speed of the block as it rotates with the turntable compared to Part I.
18. Follow the same procedure as in steps 2–10 of Part I. Students should record the data in the table for Part III. The radius will be the same as in Part I, but the time for 10 revolutions should be less than the time in Part I.

Part IV. Less Friction

19. Place the block lengthwise on an area of bare wood along the edge of the turntable so the radius is the same as in Part I (see Figure 4). Explain to students that this is simulating wet or icy roads.
20. Have students predict what effect the decreased friction will have on the maximum “safe” speed of the block as it rotates with the turntable compared to Part I.
21. Follow the same procedure as in steps 2–10 of Part I. Students should record the data in the table for Part IV. The time for 10 revolutions should be greater than the time for Part I.
22. Have students complete the data table and answer the questions on the worksheet either individually or together as a class discussion.

Student Worksheet PDF

11963_Student1.pdf

Teacher Tips

• This kit contains enough reusable materials to perform the demonstration an unlimited number of times: one PVC block,one 12"-diameter turntable, a 3" x 3" x 1" high wood ramp, and a 3" x 12" adhesive-backed sandpaper strip. All parts of this demonstration can reasonably be completed in one 45- to 50-minute class period. The calculations and questions may be completed the day after the demonstration.
• Have several students use timers. Eliminate outliers and take an average time. To ensure all timers start together, once a maximum safe speed is achieved, begin counting down out loud each time you touch the sandpaper, “Three, two, one, zero.” The students who are timing will get used to the rhythm and start the timers when you say, “Zero.” Continue counting up from zero to ten.
• Practice the demonstration several times to get a feel for how fast the turntable may rotate without the block moving from its position for each part. Achieving an exact maximum safe speed may not be possible. Students will still be able to observe the difference between an unsafe speed (as the block “skids” out of the turn) and a safe speed.
• Have students design experiments to test other variables (e.g., the slope of the banked curve, the mass of the object, the contact area between surfaces).

Sample Data

Data Table

Results Table

Sample Calculations

1. Calculate the average time for 10 revolutions for each part of the demonstration. Record the average times in the data table.
   

   (8.21 + 8.09 + 8.30)/3 = 8.20 s

2. Determine the rotational speed (revolutions per second) of the PVC block “car” for each part by dividing 10 revolutions by the average time. Record the rotational speed for each part in the results table.
   

   10 rev/8.20 s = 1.22 rev/s

3. The period of revolution is the time it takes the turntable to make one complete revolution and is equal to 1/rotational speed. Calculate the period for each part and record in the results table.
   

   1/1.22 rev/s = 0.82 s/rev

4. Determine the actual distance the car travels per revolution. Since the car travels in a circle, the distance is the same as the circumference of the circle, or 2πr, where r is the radius of the circle. Calculate the distance per revolution for each part of the demonstration (the distance should be the same for Parts I, III and IV) and record in the results table.
   

   2(3.14) (13.0 cm) = 81.6 cm

5. The linear speed is the distance the car travels per unit of time. Divide the distance per revolution by the period to determine the maximum safe linear speed of the block for each part. Record the linear speed in the results table.
   

   81.6 cm/0.82 s = 99.5 cm/s

Answers to Questions

1. Compare the effect of the decreased radius on the rotational speed of the turntable to the effect on the linear speed of the block. Explain the difference.

   As the radius decreased, the rotational speed increased and the linear speed decreased. Even though the turntable was rotating faster in Part II than in Part I, since the block was closer to the center of the circle, the distance it traveled was less than when the block was at the edge of the turntable. Therefore, the linear speed of the block decreased.

2. Use the results from Part III to explain why most NASCAR race tracks have high-banked curves.

   The maximum safe linear speed of the block increased when the curve was banked compared to the speed on the flat surface. NASCAR race tracks have high-banked curves to allow drivers to maintain a top speed.

3. A driver was traveling the posted speed limit on a rainy day. As his car entered a curve, the car started to skid. Use the results from Part IV to explain why he was “traveling too fast for conditions.”

   Centripetal force is provided by the friction between the car and the road. Friction is reduced on wet roads; therefore, the car should have been traveling below the posted speed limit for the curve in order to remain in the circular path.

4. You are part of an engineering team assigned to design a new curving on-ramp for an interstate highway. Describe the features you would include with the on-ramp in order for cars to reach the posted highway speed limit as they exit the ramp.

   The tightest part of the curve should be banked, and the radius of the curve should increase as cars approach the highway. The ramp’s exit should be straight or nearly straight, parallel to the highway.

Discussion

According to Newton’s first law, when an object is in motion, it will remain in motion unless acted upon by an unbalanced force. This means an object will travel in a straight line at a constant speed as long as no outside force is acting on it. Velocity is the rate of motion in a specified direction and acceleration is a change in an object’s velocity. Both velocity and acceleration are vector quantities in that they are based on a magnitude and a specified direction (i.e., the car is traveling north at 55 mph). Speed, on the other hand, is an object’s rate of motion. Speed is a scalar quantity and is based only on a magnitude (i.e., the car is traveling at 55 mph). Whenever an unbalanced force is acting on a mass, acceleration occurs according to Newton’s second law of motion (F = ma). Just as a force is required to change an object’s speed, a force is also required to change an object’s direction. Since velocity is a quantity of speed and direction, a change in an object’s speed, direction or both causes the object to accelerate.

Any motion in a curved path represents accelerated motion and requires a force directed toward the center of curvature of the path. Centripetal force is the “center seeking” force that causes an object to move in a circle. According to the second law of motion, acceleration is always in the same direction as the applied force. The centripetal force (F_c) acting on the mass is always directed toward the center of the circle, while the direction of its motion (v) is always perpendicular to the radius (r) of the circle (see Figure 5). In this demonstration, the centripetal force is supplied by the friction between the block and the turntable. As the block on the turntable increases speed, the centripetal force required to keep the block moving in a circle also increases.

With vector diagrams, uniform circular motion can be analyzed and an equation for centripetal force can be derived—most physics texts will have the derivation (see Equation 1).

Note that the centripetal force is proportional to the square of the velocity, implying that a doubling of speed will require four times the centripetal force to keep an object moving in a circle—exactly why speeding around a curve can be dangerous! This equation also shows that if the radius of the circle decreases (a tighter turn), the velocity must also decrease in order for the object to continue in its circular path.

On a flat curved road, the centripetal force is proportional to the perpendicular or normal force (in this case the road pushing up on car), which is equal to and in the opposite direction as the gravitational force acting on the car. When the road is banked, the normal force decreases; thus the frictional force decreases. However, the road is now pushing the car slightly toward the center of the circle, contributing to the centripetal force. The combination of the horizontal components of the normal and frictional forces results in a greater centripetal force with a banked curve than on a flat curved surface.

References

Eisenkraft, A. Active Physics: Transportation; It’s About Time: Armonk, NY, 2000.

Hewitt, P. Conceptual Physics, 3rd ed.; Scott Foresman-Addison Wesley: Menlo Park, CA, 1999.