Experiment 27. The rotational effect of a force applied at a distance r from a point rotation is equal to the product of the force component perpendicular to a line from the point rotation to the point of application of the force and the distance r.
A weight at a position other than the balancing point of the shelf-standard balance tends to rotate the balance. A weight on the other side can restore the balance. The weight required to restore the balance depends on the position at which is placed on the beam.
A force applied at an angle to the beam will have a balancing effect equal to that of the component of that force perpendicular to the beam applied at that point.
A force applied at a point on a rotating disk will have a rotational effect that depends on the force and the distance from its line of action to the axis of rotation of the disk.
We begin by balancing a 50 gram mass on one side of the beam by masses of 25, 50, 75 and 100 grams on the other side.
We now add masses of 10, 30 and 50 grams to one side of the beam, at different points, and attempt to balance them with a 100 gram mass on the other side.
We finally add masses of 10 grams and 50 grams on one side of the beam, at different points, and a mass of 30 grams on the other side, and balance these masses with a 100 gram mass.
We now add a 50 gram mass (a .49 Newton weight) to one side of the disk and balance it with a .98 Newtons weight at various points, directed at an appropriate angle at each point.
We finally balance the Styrofoam wheel with forces exerted add various positions and at various angles.
Answer the following questions:
Experiment 28. When a beam initially at rest, constrained to rotate in a horizontal plane about a fixed point, is accelerated by a horizontal force F, acting perpendicular to the beam, through displacement `ds, the angular velocity attained by the beam is proportional to the square root of F * `ds.
Associated video clip is on the CD EPS02.
A beam with a mass distribution symmetric about the center of the beam is (precariously) balanced on a low-friction plastic support. A torque is applied to the beam, initially at rest, over a short angular displacement by the gravitational force on a weight hanging over a pulley. The angular positions and clock times of three events are recorded: the instant at which the system is released, the instant at which the force/torque on the system ceases, and the instant at which the system comes to rest.
The angular velocity attained by the beam between the first two events is inferred from the angular displacement and the time required for the beam to stop, assuming a constant angular acceleration. The initial and final angular velocity and the angular acceleration between the second and third events are determined from displacement and time duration, again assuming constant angular acceleration. The results are checked for consistency between the final velocity found for the first phase and the initial velocity found for the second; consistency within the bounds of experimental error is evidence that angular acceleration is indeed constant.
The angular acceleration due to the torque is then inferred from the displacement through which the torque acted and the change in angular velocity.
By changing the distribution of masses on the beam, the relationship between torque, angular acceleration and the total mr^2 contributed by all masses is determined.
The kinetic energy changes in the system, as determined from the sum of all the .5 mv^2 contributions of all the masses comprising the system, is related to the applied torque and the angular distance through which it acts.
The beam can also be collided with another beam suspended above it in order to study conservation of angular momentum.
The precariousness of the balance provides direct and often frustrating experience of stable vs. unstable equilibrium.
The effect of the diameter of the plastic support on the acceleration of an otherwise freely rotating system provides opportunity to more deeply conceptualize the ideas of frictional force and torque.
Experiment 29. When a weight attached by a thread around the horizontal axle of a rotating disk falls, the average velocity of the falling weight is proportional to the average angular velocity of the disk.
Associated video clip is on the CD EPS02.
By moving the weight a known distance and measuring the resulting angular displacement of the disk, the ratio of the linear distance to the angular distance can be determined. The ratio of the distance moved by of a point on the disk to the distance moved by the weight can also be seen to be equal to the ratio of the distance of the point from the axle and the radius of the axle.
By measuring the time required for the weight to descend from rest to different distances below its initial point, it can be determined that the acceleration of the weight is approximately uniform. It can then be inferred that the acceleration of the disk is approximately uniform, and that the average velocity of the weight is therefore equal to the average of its initial and file velocities, with a similar relationship for angular velocities of the disk.
The proportionality constant between the inferred average velocities of the weight and average angular velocities of the disk can then be related to the radius of the axle.
Experiment 30. When a change in the angular velocity of a disk is the result of a changing gravitational or elastic potential energy, the change in the total kinetic energy of the rotating disk is equal and opposite to the potential energy change.
Associated video clip is on the CD EPS02.
Steel bolts are embedded in a Styrofoam disk, forming rings with varying numbers of bolts and various radii. The disk is constrained to rotate about horizontal axle.
By allowing a weight attached a thread wound around the horizontal axle of the disk to fall through a known distance, with the entire system initially a rest, the time required for the weight to fall permits us to determine the average velocity of the weight and to infer, from the presumed uniform acceleration, its final velocity. From this final velocity and the radius of the axle the final angular velocity of the disk can be inferred. From the velocities of the various bolts, and hence their kinetic energies, can be determined. The potential energy loss of the falling weight, the total kinetic energy of the bolts and the (negligible) kinetic energy of the falling weight can then be compared.
The relationship between the torque exerted by the falling weight, the total of the mr^2 contributions of the bolts, and the angular acceleration of the system can be determined, with the conclusion that the angular acceleration is equal to the torque divided by the total of the mr^2 contributions. This leads to the definition of moment of inertia.
Experiment 31. The harmonic motion of a pendulum or weight on a spring can be modeled with great precision by the appropriate coordinate of a point moving at constant velocity around a circle of appropriate radius.
Associated video clip is on the CD EPS02. This clip demonstrates the DOS-based program. The Windows-based program (see next figure) functions similarly but you use the mouse in a way described in the instructions given for this version.
NOTE THE NEW WIDOWS VERSION OF THE SHM PROGRAM. Click here for Revised Instructions using the Windows version.
A computer simulation of a point moving around a circle, with the horizontal and vertical projection lines of the point extended to span the entire screen, can be used to model the SHM of a pendulum or of a mass suspended from a spring. The program permits continuous user control of the radius of the circle and the angular velocity of the point (angular velocity can be refined to a precision of .001 rad/sec). From the angular velocity, which can be thus determined with excellent precision, and the mass of the object we can very accurately determine the value of the force constant or the length of the pendulum.
Using a pendulum with length between 1 meter and 2 meters, synchronize the simulation with the pendulum.
The angular velocity of the point on the reference circle (i.e., the point moving around the circle on your computer screen) should be
where k is the restoring force constant for the pendulum and m is its mass. The quantity `omega will then be in rad / sec when k is in N / m and mass m in kg.
The restoring force constant k is the constant such that F = k * x, where F is the restoring force and x is the displacement from equilibrium (remember measuring force vs. displacement with the rubber band, graphing F vs. x, fitting a straight line, and finding the slope).
Remember also that F is in the same proportion to the weight of the pendulum as displacement x to the length. This means that F / (m * g) = x / L.
So F = m g x / L = ( m g / L ) * x.
Since F = k x this means that k = m g / L.
Then change the length of your pendulum, without measuring the length, and use the simulation to accurately determine the angular frequency `omega, which is shown on the screen.
Experiment 32. The total of the potential and kinetic energies of a simple pendulum whose amplitude is much less than its length is very nearly constant.
Associated video clip is on the CD EPS02.
By measuring the restoring forces on a simple pendulum consisting of a sphere at the end of a string, at various positions between equilibrium and maximum displacement, we can obtain data from which a function modeling potential energy vs. position can be obtained.
** modify for washer on cylindrical pendulum ** By permitting the pendulum to collide at various points in its motion with an identical stationary sphere and by then determining the resulting velocity of the sphere, either by its range as a projectile or by the time required for it to travel a known distance along a constant-velocity ramp, we can determine the velocity of the pendulum mass at various points and hence its kinetic energy. This requires prior knowledge of the coefficient of restitution for the two masses, which can be determined from the type of data gathered in the conservation of momentum experiments.