Sinusoidal Nature of Simple Harmonic Motion PDF

Sinusoidal Nature of Simple Harmonic Motion Physics 2 LG 2.3 SINUSOIDAL NATURE OF SHM  SINUSOIDAL NATURE OF SHM Consider the several instances in the motion of a bungee jumper as he oscillates up and down while being held on a bungee spring. SINUSOIDAL NATURE OF SHM...

Sinusoidal Nature of Simple Harmonic Motion Physics 2 LG 2.3 SINUSOIDAL NATURE OF SHM  SINUSOIDAL NATURE OF SHM Consider the several instances in the motion of a bungee jumper as he oscillates up and down while being held on a bungee spring. SINUSOIDAL NATURE OF SHM Figure B shows the vertical position y of the bungee jumper as a function of time while Figure C shows velocity as a function of time. Notice that the velocity is maximum when the bungee jumper is at the equilibrium position and is zero when the bungee jumper is at either the highest or at the lowest point. SINUSOIDAL NATURE OF SHM Going from the equilibrium position up to the highest point, the jumper slows down until he stops and then proceeds to move downwards. The jumper speeds up as he goes back to the equilibrium position where he attains his maximum speed. SINUSOIDAL NATURE OF SHM Lower than the equilibrium position he continues moving downwards but with decreasing speed until he momentarily stops at the lowest point. He then switches direction and speeds up as he goes back to the equilibrium position. SINUSOIDAL NATURE OF SHM Both position and velocity vary sinusoidally with time, characteristic of simple harmonic motion. For the given example, with the jumper at the equilibrium position at t=0  SINUSOIDAL NATURE OF SHM Notice that the acceleration is of the same form, that is a sine function, as the position. This implies that both position and acceleration attain maximum value at the same time. The acceleration is maximum when the jumper is at maximum displacement where the spring force is also maximum. The acceleration is zero at zero displacement or at the equilibrium point where the force is zero. VERTICAL SHM VERTICAL SHM For vertical SHM, note that there are actually two forces acting on the oscillating body: the spring force and its weight. In the absence of an attached mass, as shown in Figure a, the spring hanging with its end attached to a fixed point is at its natural length. VERTICAL SHM Attaching a mass on its other end will stretch the spring to new equilibrium length where it is stretched by an amount determined by the weight of the mass as shown in Figure b. Note that at this new equilibrium position, the spring force cancels out the weight of the mass: ks = mg. VERTICAL SHM When the mass is below this new equilibrium position, as in Figure c, spring force is directed upwards and the net force on the mass is: VERTICAL SHM When the mass is above the equilibrium position and the spring is compressed from its natural length, the spring force is directed downwards and the net force on the mass is: VERTICAL SHM In both cases the net force is of the form of Hooke’s law with the force directly proportional to and opposite to the displacement y from the new equilibrium position. This means that even if the weight acts on the system, we can simplify the equation as in the case of horizontal SHM as long as we are measuring with respect to the new equilibrium position.  VERTICAL SHM Recalling Newton’s 2nd law of motion F = ma and with the net force given as F = -ky we can see that the acceleration is proportional to the displacement but opposite in direction. Plug in the sinusoidal expressions for position (which is the displacement from the equilibrium) and acceleration to the equations for force:  VERTICAL SHM Recalling Newton’s 2nd law of motion F = ma and with the net force given as F = -ky we can see that the acceleration is proportional to the displacement but opposite in direction. Plug in the sinusoidal expressions for position (which is the displacement from the equilibrium) and acceleration to the equations for force: Let’s check your understanding!  REAL -WORLD PROBLEM SOLVING: SPRING  REAL-WORLD PROBLEM SOLVING: SPRING Based on the velocity vs. time graph in the figure, which of the labeled points correspond to the following situations? a) Zero acceleration 1, 5, 9 b) Maximum acceleration 3, 7 c) Zero displacement 1, 5, 9 d) Maximum amplitude 3, 7 e) Maximum force 3, 7  REAL -WORLD PROBLEM SOLVING: SPRING  REAL-WORLD PROBLEM SOLVING: SPRING A 2.00-kg, frictionless block is attached to an ideal spring with spring constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find the amplitude. In life, there will be external forces that may cause compression or stretching within us. Embrace the resilient attitude of spring. Choose to endure and return to your relaxed state. Endure the challenges; they are temporary. Embrace the resilience that these experiences have contributed to you.