Periodic Motion, Oscillation Instructor: Xiao, Yong ( 肖湧 ) ， Wang Kai( 王凯 ) TA: Li, Yueyan （李跃岩） Recitation TA: Zhai, Chenyu （翟宸宇） General Physics

Periodic motion When an object vibrates or oscillates back and forth, over the same path, each oscillation taking the same amount of time, the motion is periodic.

Uniform Circular Motion If a body traverses a circular path at constant speed, it must be exerted by a centripetal force which is also constant in magnitude and direct at all times towards the axis of rotation. F : Centripetal force v : Speed : Unit vector along the radial direction m : Mass of the body r : Radius

Uniform Circular Motion The equation of motion: We know In physics, if the body repeats one cycle again, the angular will be increased by 2π. Thus, the period T, is defined as the time required for one complete cycle.

Uniform Circular Motion If we consider only the motion along x-axis, we have Use Newton’s Law:

Uniform Circular Motion The solution can be written in a form such as: Prove:

Oscillations of a spring Hooke’s Law When the spring is compressed, or stretched, the spring exerts a force that acts in the direction of returning the equilibrium position. Where k is a constant factor characteristic of the spring. The minus sign reminds us that the force always makes the spring return to its normal length.

Oscillations of a spring Equation of motion Using Newton’s Law Let’s make an analogy with So we have

Oscillations of a spring Dimension of k Applying dimensional analysis, we can determine the dimensions of k. From Dimensions of mass m is Dimensions of acceleration a is Dimensions of x is So the dimension of k is, the unit of k is

Oscillations of a spring Potential Energy The work done by the spring force from A to B is: We can see the work depends only on the end points, and not on the path taken, so the force is conservative force. It is usually convenient to choose the potential energy at its normal length (x=0) to be zero. So the potential energy of a spring compressed or stretched an amount x is:

Oscillations of a spring Example An object of mass m is attached to the end of a coil spring, and it oscillates on the horizontal smooth surface. The equilibrium position is at x=x0. When the spring is compressed to the point x=x 1, the speed of the object is zero. Calculate the speed when it moves to the point x=x 2.

Oscillations of a spring Using conservation of energy, we have:

The Simple Pendulum A simple pendulum consists of a small object suspended from the end of a lightweight cord. We assume that the cord doesn’t stretch and its mass can be ignored. The pendulum oscillates along a small part of a circle with equal amplitude on both sides of its equilibrium point. We have:

The Simple Pendulum Using Newton’s Law: Compare with We have and

The Simple Pendulum Considering the conservation of energy, we have Expand, when is small:

The Simple Pendulum So the energy conservation equation can be written as:

Periodic motion Equation of motion: uniform circular motion Oscillations of a spring or the simple harmonic oscillator The simple pendulum

Wave A wave is disturbance or oscillation, that travels through matter or space, accompanied by a transfer of energy. Wave motion transfers energy from one point to another, with little or no associated mass transport. Mechanical waves propagate through a medium, for example, sound waves propagate via air molecules colliding with their neighbors. Electromagnetic waves do not require a medium.

Wave Transverse or longitudinal wave: Transverse waves occur when oscillations are perpendicular to the propagation, for example electromagnetic wave. Longitudinal waves occur when the oscillations are parallel to the direction of propagation, for example sound waves.

Wave We use a transverse wave to introduce some important quantities. The high points on a wave are called crests or peaks, the low points troughs. The amplitude is the maximum height of a crest, or depth of a trough. The distance between two successive crests is called the wavelength λ.

Wave If, so when x=0,y=0 and x=λ,y=0 we have or. We can draw a picture of the oscillation of a crest. The period T, is the time required for one complete oscillation. The frequency f, is the number of crests or complete cycles per second.

Homework #4 Equation of motion P388/4 Hooke’s Law P388/2 Equation of motion P389/ Potential Energy P390/28 33 The Simple Pendulum P391/47 Wave P419/24