PURPOSE

To study uniform circular motion by determining the centripetal force required to keep a known mass moving at constant linear speed in a circular path.

EQUIPMENT manual variable speed rotator, mass holder with masses, photogate with DataLogger interface, half-meter stick, vernier caliper.

RELEVANT EQUATIONS

DISCUSSION

By definition, acceleration is a vector quantity that is proportional to the change in the velocity vector, whether the change be one of magnitude or of direction, or both. The acceleration therefore has the same direction as the change in velocity. We will call the velocity change Dv. The exact value of the acceleration is obtained through a limiting process as the time rate of change in the velocity vector when the time interval approaches zero:

                                                     (1)

Consider a particle at point P moving at uniform speed counterclockwise around a circle, as illustrated in Figure 21-1. At any point on the circle, the instantaneous velocity of P can be represented by a vector arrow tangent to the circle, and with a length proportional to the magnitude of the velocity. Since the speed of P around the circular path is the magnitude of the velocity, the velocity vector in uniform circular motion will always have the same magnitude, but its direction will be constantly changing. This results in a change in the velocity vector, and therefore an acceleration.

In order to determine this velocity change, refer to Figure 21-1, and consider the points P_o and P₁. The vector v_o represents the instantaneous velocity of P at P_o. Likewise, v₁ represents the instantaneous velocity of P at P₁. We assume that it takes an amount of time equal Dt for the point to move from P_o to P₁.

The change in velocity Dv, is calculated as follows. Since the motion is uniform, the average change in velocity will occur at the midpoint of the travel, and so we will make a construction on the extended radial bisector of arc P_oP₁. The vector change in velocity is the difference between the final and the initial velocity over this interval:

                                                    (2)

This subtraction of vectors can be done graphically. Move v₁ over so that its tail falls on the extended radius bisector. Now subtract v_ofrom v₁, to get Dv. Remember that when a vector is subtracted, it is simply flipped by 180°. In other words, the expression (v₁ - v_o) = v₁ + (-v_o). A negative vector, -v_o, is simply v_o reversed.

Make the addition by the tip-to-tail method, as shown. Note that the vector Dvis directed toward the center of the circle. Therefore the average acceleration will also be directed toward the center of the circle, because the denominator Dt in equation (1) is not a vector quantity and has no directionality associated with it.

We can conclude therefore, that whenever an object is observed to be undergoing uniform circular motion, it is being accelerated, and the vector a is directed toward the center of the circle every instant; its direction is "centripetal". The magnitude of the centripetal acceleration is given by:

                                                        (3)

where v is the uniform speed of the object and r is the radius of the circular path. This relationship is derived in your textbook for reference.

According to Newton’s Second law, any acceleration must occur in direct proportion to a net force acting on the object. In this case therefore, the force must also be directed toward the center of the circular path at every instant. The descriptive term that is used is "centripetal force". Note that "centripetal" refers only to the direction, and does not say anything about what the origin of the force is. The "centripetal" form of the Second law can be written:

                                    (4)

Normally, when an object is travelling round and round in a circle, the easiest thing to measure is the time that it takes to complete one turn, or the period T, of the motion. Since the speed is constant, and the distance covered in one turn is just the circumference of the circle, or Ds = 2 p r, the speed can be determined from the period using the following relation derived from the definition of constant speed:

                                                      (5)

Figure 21-2: Centripetal Force Apparatus

Another commonly used term is the angular speed w, as given by:

                                                          (6)

The angular speed is properly measured in units of radians/second, but other common units are revolutions/second or revolutions/minute (rpm).

The apparatus used to measure the centripetal force necessary to keep a mass whirling in a circular path at constant speed consists of a bob B suspended by string on a rod R that is attached to a rotating axle A, as shown in Figure 21-2.  The bob is attached to the axle by means of a spring that provides the centripetal force necessary for uniform circular motion. The position of the bob along the rod, and therefore its distance from the axis of rotation is adjustable. The other end of the rod has a counterweight C that can also be adjusted so as to maintain rotational stability.   The assembly can be rotated about the vertical axis by using your fingers.  The quality of the bearings is so good that it is quite easy to maintain a constant speed.

The objective is to rotate the bob at a constant speed so that its tip lines up with the pointer P, as sighted at the pointer’s tip.  The period can be readily measured by counting the number of revolutions N the bob makes in a particular elapsed time Dt.  In this case, the period is given by:

                                               (7)

The speed can be calculated from the period and a measurement of the distance of the bob from the axis of rotation.  Using this data, and knowing the mass of the bob, the centripetal force can be calculated.

The experimental test in this exercise is to compare the calculated centripetal force to the value determined by statically stretching the spring using weights W suspended over the pulley at the edge of the base.