physics lab report

Physics I: Centripetal Force Lab

Originally by Bruce Johnson, vandalized by Steven Hoke 2017

Theory: Since stuff in motion tends to stay in motion, it’s a little surprising that so much of it

sticks around. The accumulation of stuff into larger structures such as atoms, molecules,

planets, and galaxies is mostly due to centripetal forces. Centripetal comes from the Latin for

seeking a center; a centripetal force is one that continually pulls inward to a particular point, the

center. Examples of centripetal forces include gravity, the electrical attraction of outer

electrons for the protons in the center of the atom, and the tension in the string that keeps an

object whirling around in a circle.

We can determine the centripetal force required to make something move in a circle at a

constant speed – an “orbit.” For a force to change the direction of motion without changing the

speed, it must be acting perpendicular to the direction of motion; and if that force is constant

in magnitude, then it will be changing the path of motion in a constant way, resulting in a path

with constant curvature. But a path with constant curvature is just a circle, so the

perpendicular direction of the centripetal force curving that path to the side must be along the

radii of the circle. From these facts we can construct a geometric relation between the vectors

V1 , representing the velocity of the object at point A on the circle, and V2 representing the

velocity at point B.

Figure 1. Image credit http://blog.cencophysics.com/2010/02/centripetal-force/

In this diagram, the lines labeled r are radii of the circle centered at O. Δs is the change of

position between A and B, and ΔV is the change in the velocity vector, the difference ΔV = V2

– V1. Since the velocity vectors and the radii are perpendicular to each other, the radii will

sweep over the same angle as the velocity as we go from A to B. Further, since the magnitude

of the velocity is constant ( |V2|= V = |V1| ) and so is the length of the radii, both triangles

are isosceles triangles. Isosceles triangles that share the same angle are similar to each other;

that means that the proportion between respective parts of these triangles will be the same. In

particular,

∆𝑠

𝑟 =

∆𝑉

𝑉 . (Equation 1 )

We want to find force, which means finding acceleration, so we’d like to shoehorn acceleration

into this equation somehow. Acceleration can be expressed as change in velocity over time, or

𝑎𝑐 = ∆𝑉

∆𝑡 ; solving this for ∆𝑉 = 𝑎𝑐 ∆𝑡 allows us to substitute into Equation 1 to get

∆𝑠

𝑟 =

𝑎𝑐∆𝑡

𝑉 ,

which rearranges to 𝑎𝑐 = 𝑉

𝑟

∆𝑠

∆𝑡 . But

∆𝑠

∆𝑡 is another way to say V, so what we end up with is

the centripetal acceleration 𝑎𝑐 = 𝑉 2/𝑟. Given that the object has a mass M, the centripetal

force to keep it moving in a circle of radius r with velocity V is 𝐹𝑐 = 𝑀𝑉 2/𝑟.

In the case of a pendulum, the tension in the string - when the pendulum is still, and the only

force pulling on it is gravity - is increased by the centripetal force needed to keep it in the arc

of the pendulum whenever it is in motion.

Apparatus : Force probe, LabQuest Mini, string, 200 g mass, Photogate, posts, meter stick,

calipers, and a 1.000 kg or 0.500 kg mass to calibrate the force probe.

Experiment

• Attach the LabQuest Mini to the computer, if needed, and the force probe and

photogate to the LabQuest Mini. The force probe should be on the ± 10 N scale.

• Insert a vertical post into the hole in your table and attach a crossbar near the top.

Attach the photogate near the bottom of the post, under the crossbar.

• Tie a string to the crossbar and to your 200 g mass so as to create a pendulum. The

pendulum bob should swing through the center of the photogate.

• Make sure that your vertical post is pressed firmly into the hole in the table to minimize

any motion of the bar where the string is attached.

• Using a meter stick measure the radius of the pendulum – the length from where the

string is tied to the bar, to the center of the mass cylinder - and record it in Table 1 in the

worksheet.

• Using calipers, measure and record the diameter of the mass cylinder.

• Calculate the expected weight of the mass cylinder, using g = 9.80 m/s2, and record it in

Table 3 of your worksheet.

• Let the mass hang; measure its weight with the force probe, and record it.

• Compare the measured weight with the expected, using percent error and recording the

result in Table 3. If it is large, try to identify and correct the problem; repeating previous

steps as necessary.

• Set the duration of data collection to 1.0 s and the samples/s to 500.

• Pull the mass to an angle of 45º or larger and carefully release it so that it swings

through the center of the photogate. Let the mass swing back and forth at least five

times before collecting data.

• Copy the small portion of the force data from the time that the pendulum blocks the

photogate to when it unblocks it, and paste it into your worksheet in Table 2.

• Record the times when the bob blocked the photogate, and when it stopped blocking it,

in Table 2.

• Calculate how long it took for the pendulum bob to pass through the photogate, from

blocking it to unblocking it, and record it in Table 3.

• Using the diameter of the pendulum mass as the distance traveled over the time

interval above, calculate and record the speed of the pendulum during that time.

• Calculate and record the centripetal acceleration of the mass at the bottom of the swing.

• Calculate and record the centripetal force on the mass at the bottom of the swing.

• Draw a force diagram of the pendulum when it is at the bottom of the swing, passing

through the photogate; let the y axis be parallel to the string. Use the diagram to solve

for the tension in the string. Make sure that your forces are labeled and have

arrowheads clearly marked to indicate the direction of the force. After forces have been

labeled, and axes have been drawn, start applying Newton's laws by writing the line

∑ 𝐹𝑦 = 𝑚𝑎𝑦 . The next line should have the forces and acceleration filled in; then solve

for the force of tension in the string. Put the diagram into the Diagram tab of the

worksheet.

• Calculate the total theoretical force of tension in the string using the formula derived

above, and record it in Table 3.

• Calculate and record the average of the measured force in the string during the time

when the Photogate was blocked.

• Calculate a percent error between the measured tension and expected tension in the

string, and record it in Table 3.

Errors and Conclusion

In the conclusion tab, comment on any causes of inaccuracy in your measurements. List as

many as you can, including at least three sources other than yourselves or the instruments,

and any ways you can think of to correct, eliminate, or account for those errors.