Cycle I Advanced Placement Physics

John Dewey High School

Mr. Klimetz

Centripetal Force

Method and Apparatus for the Demonstration and Calculation

of Centripetal Force

How can we determine the centripetal force acting on an object traveling in a circular path?

How can we establish the relationship between the centripetal force exerted on an object traveling in a circular path

and its orbital period, orbital frequency, mass, orbital velocity, orbital radius, and force of gravity?

Laboratory Exercise No. 5

Apparatus

Data Gathering and Calculations.

a.  Convert the six suspended mass values into kilograms and enter into data table.

b.  Calculate the weights (wt) of each of the six suspended mass values and enter into data table.

c.  Along with your lab partners, use the stopwatch to measure the time it takes to complete 30 full orbits of the rubber stopper. Repeat this three times for each mass increment and average. Enter into data table.

d.  Calculate the average orbital period (T) for each mass increment by dividing the average of 30 full orbits obtained in Step C above by 30. Enter into data table.

e.  Calculate the orbital frequency (f) for each mass increment by calculating the inverse of the average orbital period, that is, by setting f = 1/T. Enter into data table.

f.  Calculate the orbital velocity (v) for the given orbital radius and period by setting v = 2pr/T, where p = 3.14. Enter into data table.

g.  Calculate the centripetal force (Fc) from the relationship Fc = mvexp2/r. Enter into data table.

h.  Compare the vertical force (wt) of each suspended mass increment to the calculated centripetal force (Fc).

Questions

1.   Plot a graph of orbital velocity (v) versus orbital period (T)  for each of the six individual values of applied centripetal force. Period should be regarded as the independent variable and orbital velocity should be regarded as the dependent variable. Make sure to set-up your axes properly with each commencing numeration at zero. Create a separate table of values of v versus T before plotting your data.

2.   Based on your graph of v versus T, briefly explain the observed relationship between orbital velocity and orbital period of an object traveling in a circular path at a constant orbital radius.

3.   Assuming that the Moon orbits the Earth in a circular path, calculate the Moon's orbital velocity (vm) and orbital period (Tm) given the following data and equations from the Reference Tables for Physics:

Mean distance from Earth to Moon: (r) = 3.8 x 10exp8 m

Mass of the Earth (me) = 6.0 x 10exp24 kg

Mass of the Moon (mm) = 7.4 x 10exp22 kg

Universal Gravitational Constant (G) = 6.7 x 10exp-11 N-mexp2/kgexp2

Force of Gravitational Attraction between Earth and Moon (Fg) = G x me x mm/rexp2

Centripetal Force Exerted on Moon by Earth (Fc) = mm x (vm)exp2/r

Orbital Velocity of the Moon (vm) = 2pr/Tm

Orbital Period of the Moon (Tm) = 2pr/vm

[Hint: Set Fg = Fc and solve for vm. Then solve for Tm.]

Express vm in km/s and Tm in days. How is the length of time Tm more familiarly known?

Introduction. As long as no resultant (net) force acts obliquely upon an object in motion it will travel along a linear path. If we wish to make an object travel in a circular path, however, a force acting towards the center of the circle must be applied to the object. This force is called a centripetal force (Fc). The centripetal force must change with the orbital velocity of the object if we are to keep the object traveling along a circular path of fixed radius. The orbital velocity (v), in turn, is proportional to the frequency of revolution (f) of the object and is inversely proportional to the period of revolution (T) of the object. Similarly, the centripetal acceleration is proportional to the square of the orbital velocity and inversely proportional to the orbital radius (r). Finally, following Newton's Second Law of Motion, the centripetal force acting on an object moving along a circular path is proportional to both the mass of the object (m) and its centripetal acceleration (ac). This is equivalent to the force of gravitational attraction (Fg) between itself and a central mass, much like that which exists between a planet and the Sun. This is simulated, in this particular situation, through the use of suspended masses attached to a line connected in turn to the orbital mass. This exercise is designed to illustrate the intriguing physical and mathematical relationships that exist between the centripetal force acting on an object traveling in a circular path and its period, frequency, mass, velocity, radius of travel, and force of gravity.

Cylindrical plastic ballpoint pen casing

1.5 m length of narrow gauge braided nylon cord

00 size single vent rubber stopper with a mass of 0.0045 kg

Adhesive tape

Paper clip

Meter stick

Stopwatch

Set of slotted "coin" type masses from Hooke's Law Apparatus:

1-10 gram mass

2-20 gram masses

2-50 gram masses

1-100 gram mass

Procedure. Thread one end of the nylon cord through the rubber stopper vent hole and securely attach with adhesive tape. Thread the other end of the cord through the ballpoint pen casing and tie a 4-cm diameter loop at the end. This loop will serve to connect the series of slotted masses to the apparatus. Attach the paper clip between the looped end of the cord and the bottom of the pen casing so as to establish a 0.50 m orbital radius (r) cord length between the top of the pen casing and the center of the rubber stopper. Add the masses according to the schedule listed below:

50 grams

100 grams

120 grams

150 grams

180 grams

200 grams

For each orbital velocity, the orbital radius (r) of the stopper will assume a constant value (0.50 m) and horizontal orientation, and the suspended masses will occupy a corresponding vertical orientation. Whirl the stopper over your head in a horizontal circular path at such a rate that the paper clip is just pulled up to, but not touching, the bottom of the pen casing. Here, the orbital radius of the rubber stopper should be exactly 0.50 m. Reposition the paper clip on the cord accordingly in order to ensure the 0.50 m orbital radius. Adjust the orbital velocity (faster v or slower v) so that the radius of the circular path remains at the constant 0.50 m value. This is accomplished by the "limiting" paper clip attached to the cord so that the masses remain suspended a constant distance below the bottom end of the pen casing. The centripetal force (Fc) furnished by the weight (wt) of the suspended masses is now keeping the stopper moving in a circular path of fixed radius.

It is important to keep two factors constant during the course of this exercise: the mass of the orbiting object and the radius of the orbit. The centripetal force is varied by altering the amount of mass suspended from the string. Note that the mass of the rubber stopper is 0.0045 kg and the mass of the cord is neglected.