Periodic Motion

Periodic motion is any motion that repeats itself, e.g., a spinning carousel, a yo-yo, a pendulum in a clock. Recall that a circle is made by joining together an infinite number of infinitely small straight lines (chords), each at a slight angle to its neighbours. And, because the circle is made of these straight lines, we can describe circular motion with equations usually used with straight line motion. Recall that we previously used straight line motion equations with curving motion when we studied arcs by separating the arc motion into its two straight line horizontal and vertical components.

Bending Straight Line Motion              If you are swinging a rock on a string around your head and the string breaks, in which direction does the rock take off? In a curving path? a straight line path? A common guess is straight out from the center of the circle (along a radius line). The answer, perhaps surprising, is straight ahead, on a tangent from the point at which the string breaks. Think: the pitcher in baseball essentially moves their arm in a circle and they release their grip on the ball when their arm is more or less overhead. Where does the ball go? Straight ahead, toward the plate, on a tangent to the circle. Now, a similar situation but on a huge scale – satellites or space craft in orbit around the Earth. Why do they not just fly off into space? Gravity. Without the pull of gravity they would move away from the Earth in a tangent, just like the ball leaving the pitcher’s hand after she releases her grip. For a simple model of the effect of gravity on the path of satellites orbiting the Earth, hold a tennis ball (Earth) and slip a straight piece of piece of thin wire about 5 cm long under your index finger. Because the wire contacts the surface of the tennis ball at just one point, where your finger is, it is a tangent and shows the path a satellite would take if there were no gravity. Now, since gravity exists and pulls toward the center of the Earth, with your other hand, push the end of the wire toward the tennis ball. As the wire bends, it shows the effect of gravity on the tangential path of the satellite. In detail, the satellite does move out away from the Earth on its tangential path but gravity bends the path inward toward the Earth. Now the satellite starts moving out on a slightly different tangential path that gravity bends instantly back toward the Earth. Again the satellite starts moving out on yet another slightly different tangential path that gravity bends instantly back toward the Earth. This is a constant two phase process: i) the satellite moves out away a tiny bit from the Earth on a tangential path ii) gravity quickly pulls it back toward the Earth. The result: the satellite’s “out – in” motion becomes a smooth orbital path around the Earth. Look at a circular saw blade. It is a circle made of numerous “out and in” shapes, the teeth.

             The proper name for the acceleration of gravity in this situation (holding objects in orbit by pulling in on their tangential escape paths) is centripetal acceleration, a name created by Newton, so famous for the study of light and motion. The force creating the acceleration is the centripetal force.

             Is gravity the only thing that creates the centripetal force? It depends on the situation. If you want to keep a rock on a string spinning around your head, you must use your arm muscles to create the necessary inward pull. Consider an extreme example of this type, the “hammer”. The hammer is metal rod or a chain with a heavy metal ball attached to one end and a hand grip at the other end. An athlete grips the hammer and begins to spin around and around, faster and faster. The effort needed to keep the hammer from flying away is enormous – the athlete’s arm muscles bulge and they arch their whole body backwards to keep the hammer moving in a circle. Finally, when the athlete has reached their maximum orbital speed, they release their grip and down the field flies the hammer, on its tangential path. If the situation is a satellite that is to stay in an orbit around the Earth, gravity must exert an inward pull on it. If carousel animals are to stay in their place as the carousel spins around, the wood planks of the platform must be strong enough to exert the required inward pull. A thing about forces is that we take them for granted; we may not notice them or be aware of what creates them until something breaks and objects go flying.

Circular Motion           When objects moves in circles, we can describe the centripetal acceleration that hold them in their circles, their orbital velocities, the diameters of the circles, and the periods of the orbits.

1. Centripetal acceleration. Find the centripetal acceleration needed to keep a carousel animal fixed to the

    18 m diameter platform as it spins around at 1.3 m/s. Use ac = v² / r

    ac = v² / r

            (1.3 m/s)² / 9 m = 0.19 m/s²

2. The period of an orbit. Find the period of a satellite orbiting the Earth at an altitude of 500 km and

    moving at 700 m/s. Use T = 2πr / v. Note: radius of earth is 6.37 * 10 ⁶ m; 1 km = 1 000 m or 10 ³ m

    T = 2πr / v

           2π ( 6.37 * 10 ⁶ m + 5 * 10 ⁵ m) / 700 m/s

           2π (6.87 * 10 ⁶ m) / 700 m/s = 6.16 * 10 ⁴ s

3. Centripetal acceleration. By substituting 2. into 1. we get ac = 4π ² r / T ² Find the centripetal

    acceleration holding a satellite in a 400 km orbit around the Earth if the orbit’s period is 5 000 s.

    ac = 4π ² r / T ²

            4π ² ( 6.37 * 10 ⁶ m + 4 * 10 ⁵ m) / (5 000 s) ²

            4π ² (6.77 * 10 ⁶ m) / 2.5 * 10 ⁷ s ² = 10. 69 m/s ²

4. Centripetal Force. Recalling that F = ma, we can just include the object’s mass in 1. or 4. to find the

    force creating the acceleration.

    a) Find the centripetal force keeping a 20 kg mass moving at 5.2 m/s around a 12 m diameter circle.

        Use Fc = m ac = m v² / r

                                   20 kg * (5.2 m/s) ² / 6 m = 90.13 N

    b) Find the centripetal force keeping a 450 g ice chunk orbiting the Earth at an altitude of 750 km and

        completing an orbit in 1.3 hr.

        Fc = m ac = m 4π ² r / T ²

                             0.450 kg 4π ² ( 6.37 * 10 ⁶ m + 7.5 * 10 ⁵ m) / (1.3 hr * 3 600 s/hr) ²

                             0.450 kg 4π ² (7.12 * 10 ⁶ m) / (4680 s) ²

                             0.450 kg 4π ² (7.12 * 10 ⁶ m) / 2.19 * 10 ⁷ s ² = 5.77 N

Torques            We know a centripetal acceleration is needed to keep an object moving in a circle and, since it is directed toward the center of the circle, this type of acceleration affects the direction of the velocity, i.e., bends a tangential velocity into an orbit. But, because of the inward (perpendicular) direction of the acceleration, it has no effect on the size of the velocity, no effect on how quickly the object moves. So, what force gets an orbiting object moving faster or slower? The force must be in the same direction as the motion, parallel to it, like a tangent, and not perpendicular, like the centripetal force. Such a force is called a torque. Think about opening or closing a door. How is this motion like a satellite or a carousel? Simple: if not for the door frame to stop it, the door panel would swing in a circle around its hinge, just like a satellite moves around the Earth. So, the door and the satellite and the carousel are related because they move in the same type of path – a circle around a center of rotation. In what direction do you apply a force to move the door? Inward, toward the hinge, like a centripetal force? No, you push or pull on the door in the same direction you want it to move – you create a torque. A last point about torques: the point of application is important. Try to close a door with a single finger next to the knob. Now try it with your finger next to the hinges. See that a force applied far from the pivot point (at the edge of the door) is more effective than one applied close to it (by the hinge).

6. What torque is created by applying a 12 N force to the door 30 cm from the hinges? Use: T = f d

    T = f d

           12 N * 0.30 m = 3.6 Nm

7. What torque is created on a rusted nut by applying a 15 N force halfway along a 20 cm long wrench?

    T = f d

           15 N * 0.10 m = 1.5 Nm. Note: a torque wrench is a type of wrench that measures the torque

           being applied to nuts and bolts to prevent their being warped by too much force.

Simple Harmonic Motion         Before we understand simple harmonic motion, we must look first at

vibrational motion, motion that occurs when something repeatedly moves in a back and forth way. Some common examples are a swing in a playground, a pendulum in a Grandfather clock, or a vibrating violin string. A less familiar example would be a weight bouncing up and down suspended on a spring. Note that in each example there is some force acting on the object to try to bring it back to its rest or equilibrium position but the object’s momentum carries it past the rest position. Again, a restoring force tugs at it; its motion slows, it stops and then back it comes toward the rest position. The amount of motion (and the amount of the displacement to either side of the rest position) gradually lessens due to the inexorable tug from the restoring force, friction at the pivot point where the object is attached to its support and friction within the vibrating material itself.

             What is the restoring force in these examples? It depends on the situation: gravity is the force that tries to keep the swing and the pendulum from moving; the elasticity in the violin string and the spring tries to keep them at their rest positions.

             In each of these situations, note that the magnitude of restoring force depends on the displacement of the object, i.e., it is largest when the object is farthest from its rest position. Get a thick elastic band. The more you stretch it (make it longer than its rest length), the more its restoring force pulls back against you. Now, get a spring made of thick metal and feel that the more you squish it (make it shorter than its rest length), the more its restoring force pushes back at your fingers. (Insta-review: The elastic or spring acting against your force reminds us of which of Newton’s Motion Laws? The 3^rd law, because it states that for every action, there is an equal and opposite one.) Put a friend on a swing and pull them back just a tiny bit. This is easy to do because you do not move them much away from the rest point so there is much not much restoring force (and so not much motion to and fro). Now pull them back about two meters: much harder to do because the restoring force is much larger and so is the to and fro motion. (Now is not the time to be standing in front of the swing!) Think of it this way: pulling the person back more lifts them higher from the ground — their potential energy increases more — so when you let them go, their speed and kinetic energy will be more than when you pull them back just a bit.

             Now a bit of review about how the restoring force affects the object’s acceleration and speed. Think of the weight bouncing up and down, suspended on the spring. Pull down on the weight and the distortion of the spring creates a large restoring force. We know from Newton’s first Motion Law that a large force acting on a mass gives it a large change in speed. But, recall that, unless the object can move for a sufficient distance and time, even a large restoring force will have minimal effect. When we hold down the weight on the spring we can feel the large restoring force but, until we release the weight, nothing happens of course.

             When is the object moving most quickly? most slowly? Make a pendulum from a brick tied to a rope supported from the ceiling. Well, of course, the brick has no speed when you are holding it but once it has started its vibrational motions, its speed changes constantly. By the time the brick swings through its rest position, the restoring force has had the maximum time and distance to increase the brick’s speed to its maximum. It does move farther as its momentum carries it beyond the rest position but, but by then the restoring force is slowing the brick until, at the other end of its travel, it is momentarily motionless. The cycle begins anew as the restoring force then begins to move the brick faster and faster (and then slower and slower) back toward you. Note: it is not correct to say that a force “creates” an acceleration nor that an acceleration “causes” the speed to change. The force acting on an object creates a speed change which is restated as an object’s acceleration value. Sometimes one might use the terms force and acceleration interchangeably but the correct relationship is: force, time and distance, speed change, acceleration.

             Finally, what is simple harmonic motion? It is the vibrational motion of an object controlled by a restoring force whose size depends directly on the displacement of the object from its rest position.

8. Find the period of a 50 cm long, 2 kg pendulum in a Grandfather clock. Recall the period is the time

    for the completion of one cycle, e.g., side - center - other side - center - side. Note: the size of the mass

    is unimportant because gravity affects all masses essentially the same. Use: T = 2π √(ℓ /g) Note: for

    two pendulums of the same length, the period is the same regardless of how far they are pulled to the

    side. How can this be? If the pendulum is pulled just a bit to the side, it does not have a chance to build

    up much speed but it does not have far to travel. If it is pulled quite far to the side, it will have farther

    to travel but its speed will be greater so the travel time will be the same. This idea is similar to the fact,

    for an object tossed to any height up into the air, the ascent time equals the descent time.

    T = 2π √(ℓ /g)

          2π √(0.50 m / 9.8 m/s²) = 1.42 s

By the way, giving a swing just a tiny push at just the correct moment will soon increase the amplitude of its vibrational (to and fro) motion considerably. The best moment to apply the force is just is as the swing is beginning a cycle. This is similar to the effect of constructive interference. The process of small energy additions combining to create a large effect is called mechanical resonance. Another example is a group of people walking in step (in phase) across a footbridge; the energy from all their footfalls combines to create a large shaking effect, sometimes strong enough to endanger the integrity of the bridge. Still another example is the failure of structures due to the energy accumulation from repeated small wind gusts. The wind gets the structure swaying a bit and then, if mechanical resonance occurs, the energy from repeated small gusts accumulates and can cause so much distortion in the structural elements, they pull apart. The disintegration of the Tacoma Narrows Bridge is a well known example of this particular phenomenon.