Recall that the gravitational force acts at the center of mass of a rigid-body. Here, the distance from the pivot to the center of mass is L. Apply the small-angle approximation, sin (t) ≈ 0(t) Note that this analysis will only provide an accurate predictive model when the angle 0(t) is small. From this you can determine the equation of motion for the system d² MgL ·O(t) = ·0(t) (6.10.1) dt² I The equation of motion is in the form of equation (6.5.1) and so you can directly determine the oscillation frequency and period of the system. MgL 2πT ω (6.10.2) I T This equation is general for any rigid-body object so long as you specify the moment of inertia and properly determine the distance from the pivot to the center of mass. The damping constant, b, depends on the shape of the object and the viscosity of the air (or other medium) in which it moves. The damping constant plays the same role as the coefficient of friction. The negative sign reminds you that the force always opposes the velocity of the object. FD m บ k اسنس + хо -A 0 x A Figure 6.11.2. A mass on a spring including drag forces. Again, apply the general physics problem- solving strategy. First, draw a pictorial representation of the situation and choose a coordinate system. Applying the Law of Motion, you find = -kx - bvx max = Using the relationship between velocity acceleration and position, rewrite the equation in terms of position only. d² dt² x(t) + b d k -x(t) + m dt x(t) = 0 m The models you have built for simple harmonic motion of spring, simple pendulums, and rigid body oscillators have not included dissipative forces such as friction. You know from experience that a pendulum left to oscillate will lose energy. The amplitude decreases in time until all energy has been dissipated and the system comes to rest. An oscillation whose amplitude decreases in time is called a damped oscillation. Energy dissipation can be caused by dissipative forces such as air resistance and friction. heeeeeeee! Figure 6.11.1. A mass on a spring including dissipative forces. You can now analyze a spring system and consider the effects of air drag, assuming a drag force that is directly proportional to the velocity. Fdrag = -bu A pendulum does not need to be a simple mass on a string. In fact, many objects that can be modeled in the rigid-body model can act as pendulums. For example, as you walk, your leg swings like a pendulum. Consider a pendulum made from a rigid-body object. L T Center of mass 79. Fa M Distance from pivot to center of mass Figure 6.10.1. A rigid-body pendulum. Use a torque analysis and your model of simple harmonic motion to analyze the system. The gravitational torque is T = -LMg sin (t) = Ia Recall that the gravitational force acts at the center of mass of a rigid-body. Here, the distance from the pivot to the center of mass is L. Apply the small-angle approximation, sin(t) ≈ 0(t) Note that this analysis will only provide an This is the equation of motion of a damped mass on a spring. Since you know that the amplitude decreases in time, you can guess a solution of the form: x(t) = Ae-bt/m cos (wt + 0) (6.11.2 The oscillation frequency is given by, k b2 2πT W3 + (6.11.3) m 4m T You can see that the oscillation frequency is equivalent to the original oscillation frequency modified by the quantity 62/4m². Below is a graph of the position as a function of time for the damped oscillator. The cosine function modified by the exponential decay is shown. You can see that the envelope of the amplitude decays in time. The maximum displacement away from equilibrium decreases according to the function xmax(t) = A exp-bt/2m x(t) A -A The maximum displacement decays exponentially over time xmax(t) = Ae-bt/2m T 2T 37 t