28 Oscillations: The Simple Pendulum, Energy in Simple Harmonic Motion

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1 Chapter 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion Startin with the pendulum bob at its hihest position on one side, the period of oscillations is the time it takes for the bob to swin all the way to its hihest position on the other side and back aain. Don t foret that part about and back aain. By definition, a simple pendulum consists of a particle of mass m suspended by a massless unstretchable strin of lenth in a reion of space in which there is a uniform constant ravitational field, e.. near the surface of the earth. The suspended particle is called the pendulum bob. Here we discuss the motion of the bob. While the results to be revealed here are most precise for the case of a point particle, they are ood as lon as the lenth of the pendulum (from the fixed top end of the strin to the center of mass of the bob is lare compared to a characteristic dimension (such as the diameter if the bob is a sphere or the ede lenth if it is a cube of the bob. (Usin a pendulum bob whose diameter is 10% of the lenth of the pendulum (as opposed to a point particle introduces a 0.05% error. You have to make the diameter of the bob 45% of the pendulum lenth to et the error up to 1%. f you pull the pendulum bob to one side and release it, you find that it swins back and forth. t oscillates. At this point, you don t know whether or not the bob underoes simple harmonic motion, but you certainly know that it oscillates. To find out if it underoes simple harmonic motion, all we have to do is to determine whether its acceleration is a neative constant times its position. Because the bob moves on an arc rather than a line, it is easier to analyze the motion usin anular variables. θ m The bob moves on the lower part of a vertical circle that is centered at the fixed upper end of the strin. We ll position ourselves such that we are viewin the circle, face on, and adopt a coordinate system, based on our point of view, which has the reference direction straiht 197

2 Chapter 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion downward, and for which positive anles are measured counterclockwise from the reference direction. Referrin to the diaram above, we now draw a pseudo free-body diaram (the kind we use when dealin with torque for the strin-plus-bob system. O r = sinθ + a θ m F = m We consider the counterclockwise direction to be the positive direction for all the rotational motion variables. Applyin Newton s nd aw for Rotational Motion, yields: a o< = τ o< a = m sinθ Next we implement the small anle approximation. Doin so means our result is approximate and that the smaller the imum anle achieved durin the oscillations, the better the approximation. Accordin to the small anle approximation, with it understood that θ must be in radians, sin θ θ. Substitutin this into our expression for a, we obtain: a = mθ Here comes the part where we treat the bob as a point particle. The moment of inertia of a point particle, with respect to an axis that is a distance away, is iven by = m. Substitutin this into our expression for a we arrive at: 198

3 Chapter 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion m a = θ m Somethin profound occurs in our simplification of this equation. The masses cancel out. The mass that determines the drivin force behind the motion of the pendulum (the ravitational force F = m in the numerator, is exactly canceled by the inertial mass of the bob in the denominator. The motion of the bob does not depend on the mass of the bob! Simplifyin the expression for a yields: a = θ d θ Recallin that a, we have: dt d θ = θ (8-1 dt Hey, this is the simple harmonic motion equation, which, in eneric form, appears as d x = constant x (equation 7-14 in which the constant can be equated to ( π f where f dt is the frequency of oscillations. The position variable in our equation may not be x, but we still have the second derivative of the position variable bein equal to the neative of a constant times the position variable itself. That bein the case, number 1: we do have simple harmonic motion, and number : the constant must be equal to ( π f. = ( πf Solvin this for f, we find that the frequency of oscillations of a simple pendulum is iven by 1 f = (8- π Aain we call your attention to the fact that the frequency does not depend on the mass of the bob! 1 T = as in the case of the block on a sprin. This relation between T and f is a definition that f applies to any oscillatory motion (even if the motion is not simple harmonic motion. 199

4 Chapter 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion All the other formulas for the simple pendulum can be transcribed from the results for the block on a sprin by writin Thus, θ in place of x, w in place of v, and a in place of a. θ = θ cos(πf (8-3 w = w sin(πf (8-4 a = a cos(πf (8-5 w = ( θ (8-6 πf πf a = ( θ (8-7 Enery Considerations in Simple Harmonic Motion et s return our attention to the block on a sprin. A person pulls the block out away from the wall a distance x from the equilibrium position, and releases the block from rest. At that instant, before the block picks up any speed at all, (but when the person is no loner affectin the motion of the block the block has a certain amount of enery E. And since we are dealin with an ideal system (no friction, no air resistance the system has that same amount of enery from then on. n eneral, while the block is oscillatin, the enery 1 1 is partly kinetic enery K = mv and partly sprin potential enery U = kx. The amount of each varies, but the total remains the same. At time 0, the K in is zero since the velocity of the block is zero. So, at time 0: E = U E = 1 k x An endpoint in the motion of the block is a particularly easy position at which to calculate the total enery since all of it is potential enery. As the sprin contracts, pullin the block toward the wall, the speed of the block increases so, the 1 kinetic enery increases while the potential enery U = kx decreases because the sprin becomes less and less stretched. On its way toward the equilibrium position, the system has both kinetic and potential enery 00

5 Chapter 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion with the kinetic enery K increasin and the potential enery U decreasin. Eventually the block reaches the equilibrium position. For an instant, the sprin is neither stretched nor compressed and hence it has no potential enery stored in it. All the enery (the same total that we started 1 with is in the form of kinetic enery, K = mv. 0 E = K The block keeps on movin. t overshoots the equilibrium position and starts compressin the sprin. As it compresses the sprin, it slows down. Kinetic enery is bein converted into sprin potential enery. As the block continues to move toward the wall, the ever-the-same value of total enery represents a combination of kinetic enery and potential enery with the kinetic enery decreasin and the potential enery increasin. Eventually, at its closest point of approach to the wall, its imum displacement in the x direction from its equilibrium position, at its turnin point, the block, just for an instant has a velocity of zero. At that instant, the kinetic enery is zero and the potential enery is at its imum value: 0 E = U Then the block starts movin out away from the wall. ts kinetic enery increases as its potential enery decreases until it aain arrives at the equilibrium position. At that point, by definition, the sprin is neither stretched nor compressed so the potential enery is zero. All the enery is in the form of kinetic enery. Because of its inertia, the block continues past the equilibrium position, stretchin the sprin and slowin down as the kinetic enery decreases while, at the same rate, the potential enery increases. Eventually, the block is at its startin point, aain just for an instant, at rest, with no kinetic enery. The total enery is the same total as it has been throuhout the oscillatory motion. At that instant, the total enery is all in the form of potential enery. The conversion of enery, back and forth between the kinetic enery of the block and the potential enery stored in the sprin, repeats itself over and over aain as lon as the block continues to oscillate (with and this is indeed an idealization no loss of mechanical enery. A similar description, in terms of enery, can be iven for the motion of an ideal (no air resistance, completely unstretchable strin simple pendulum. The potential enery, in the case of the simple pendulum, is in the form of ravitational potential enery U = my rather than sprin potential enery. The one value of total enery that the pendulum has throuhout its oscillations is all potential enery at the endpoints of the oscillations, all kinetic enery at the midpoint, and a mix of potential and kinetic enery at locations in between. 01