Question

Lab 4: Introduction & Instructions

Centripetal Acceleration

Introduction

Velocity is a vector with both a magnitude and a direction. Since acceleration is a measure of a change in velocity over time, it seems reasonable that either the magnitude of the velocity vector could be changing, or the direction, or both. If magnitude is changing only, then the motion occurs in one dimension and the principles of algebra can be applied to the equations of motion. But suppose the opposite case was true: suppose the direction of the velocity vector was changing but the speed of the object was constant? This activity will use graphical vector analysis to demonstrate that these conditions result in the object moving in a circular path with an acceleration vector pointing toward the center of the circle. Measurement of a paper-and-pencil model of this type of motion will also let us deduce the correct mathematical relationship between the acceleration, the speed of the object, and the radius of the circular path.

Required Materials and Equipment

• Two copies of the large circular arc and one copy of the small circular arc.
• A protractor and ruler

Instructions

Download and print out two copies of the large circular arc and one copy of the small circular arc. The center of each arc is identified as a point on the paper. The radius of the larger circle is exactly twice that of the smaller circle: 9.0 cm and 4.5 cm. Measure them to verify this.

This activity imagines a point-like object traveling at a constant speed clockwise around the circular path. It can be shown from calculus that the velocity vector for an object moving along a curved arc must always be tangent to the path. It can also be shown from Euclidian geometry that the tangent to a circle is perpendicular to its radius. Graphicalanalysis will be used to estimate the acceleration of the object at the identified point under different sets of parameters, and the results will be compared to each other.

Begin with the larger circular arc. First, mark a point on the arc vertically above the center of the circle. We will analyze the acceleration of an object at this point of its motion.

Imagine the initial time starts at a 20 degree angle before the dot and the final time ends at a 20 degree angle after the dot. If we assume the initial time is  Δt/2, then the final time is also Δt/2 and Δt = t_f – t_i. Therefore, the acceleration about the point at the top of the circle (halfway between the two identified points) will be

a = Δv/ Δt = (v_f – v_i)/ Δt

The numerical measurement of this change in time is arbitrary, as long as it is assumed to be the same for all trials.

FIRST CASE: LARGE CIRCLE, BASE SPEED

Draw the radius from the center of the circular arc to the point identified as t_i. Use a protractor to identify the perpendicular to the radius through that point and draw an arrow 3.0 cm long with its tail at the mark and pointing to the right. Include an arrow head at its tip. Repeat these steps at the mark to the right of the point, as shown in the figure.

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Subtract the initial velocity vector from the final velocity vector, as shown in the next figure. Do this by ‘sliding’ the initial velocity vector to the location of the final velocity vector and making their tails touch. Measure and record the length of the change in velocity vector, Δv, in the data table.

SECOND CASE: LARGE CIRCLE, TWICE THE SPEED

Now consider an object moving twice as fast: the length of the arrow will be twice as long, but it will also travel twice as far in the same time, Δt. Therefore, repeat these steps on the second large circular arc, only draw arrows 6.0 cm long perpendicular to the radius at 40 degrees on either side of the dot.

Measure and record the length of the change in velocity vector, Δv, in the data table.

THIRD CASE: SMALL CIRCLE, ORIGINAL SPEED

Finally, consider an object moving at the original speed (3.0 cm arrow) but along a circular path of half the radius. Because the linear speed is measured as the distance along the arc over time, the correct location on the circular path for the same change in time Δt must begin and end at the 40 degree marks for the smaller circle. Therefore, draw 3.0 cm arrows at +/- 40 degree marks and add the two vector arrows graphically, as before. Measure and record the length of the change in velocity vector, Δv, in the data table.

Data Analysis

A graphical analysis has already been performed and the lengths of the three Δv vectors have already been recorded in the data table on the Report Sheet. Since the time, Δt,was the same for all three figures, then the lengths of the three Δv vectors must be proportional to each acceleration magnitude, and pointing in the same direction as the centripetal acceleration.

Questions and Conclusion

1. How does your graphical vector additions suggest that the direction of the acceleration vector is toward the center of the circle?
2. Let the length of the first Δv be the standard of comparison. By what factor does the length change when the speed is doubled? What mathematical relationship does this suggest between acceleration and speed?
3. By what factor does the length of the Δv vector change when the radius is halved? What is the mathematical relationship does this suggest between acceleration and speed?
4. Write the relationship between acceleration, speed and radius for an object moving at a constant speed along a circular path, based on the results of the data you collected.

Report Sheet

CENTRIPETAL ACCELERATION

Data

Radius (cm)	Speed (cm)	ΔV (cm)
9.0	3.0
9.0	6.0
4.5	3.0

Questions

1. How does your graphical vector additions suggest that the direction of the acceleration vector is toward the center of the circle?

2. Let the length of the first Δv be the standard of comparison. By what factor does the length change when the speed is doubled? What mathematical relationship does this suggest between acceleration and speed?

3. By what factor does the length of the Δv vector change when the radius is halved? What is the mathematical relationship does this suggest between acceleration and speed?

4. Write the relationship between acceleration, speed and radius for an object moving at a constant speed along a circular path, based on the results of the data you collected.

Answer 1

Data

Radius (cm)	Speed (cm)	ΔV (cm)
9.0	3.0
9.0	6.0
4.5	3.0