Physics 111: Mechanics Lecture 14

Dale Gary

NJIT Physics Department

Life after Phys 111 The course material of Phys 111 has given you a taste of a wide

range of topics which are available to you as a student. Prerequisite is Phys 121 or Phys 121H.

For those of you who have an interest in gravitation/astronomy, I suggest the following electives:

Phys 320, 321 – Astronomy and Astrophysics I and II Phys 322 – Observational Astronomy

For those of you interested in the biological or BME/medical aspects, I suggest the following electives:

Phys 350 – Biophysics I, Phys 451 - Biophysics II

For those of your interested in light, optics, and photonics, I suggest the following elective which Federici will be teaching this fall and Fall 2014:

OPSE 301 – Introduction to Optical Science and Engineering

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Oscillatory Motion Periodic motion Spring-mass system Differential equation

of motion Simple Harmonic

Motion (SHM) Energy of SHM Pendulum Torsional Pendulum

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Periodic Motion

Periodic motion is a motion that regularly returns to a given position after a fixed time interval.

A particular type of periodic motion is “simple harmonic motion,” which arises when the force acting on an object is proportional to the position of the object about some equilibrium position.

The motion of an object connected to a spring is a good example.

Recall Hooke’s Law

Hooke’s Law states Fs = kx Fs is the restoring force.

It is always directed toward the equilibrium position.

Therefore, it is always opposite the displacement from equilibrium.

k is the force (spring) constant. x is the displacement.

What is the restoring force for a surface water wave?

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Restoring Force and the Spring Mass System

In a, the block is displaced to the right of x = 0.

The position is positive. The restoring force is directed to the left (negative).

In b, the block is at the equilibrium position.

x = 0 The spring is neither stretched nor

compressed. The force is 0.

In c, the block is displaced to the left of x = 0.

The position is negative. The restoring force is directed to the right (positive).

Differential Equation of Motion

Using F = ma for the spring, we have But recall that acceleration is the second derivative

of the position:

So this simple force equation is an example of a differential equation,

An object moves in simple harmonic motion whenever its acceleration is proportional to its position and has the opposite sign to the displacement from equilibrium.

ma kx

2

2

d xa

dt

2 2

2 2 or

d x d x km kx x

dt dt m

Acceleration Note that the acceleration is NOT constant,

unlike our earlier kinematic equations. If the block is released from some position x = A,

then the initial acceleration is – kA/m, but as it passes through 0 the acceleration falls to zero.

It only continues past its equilibrium point because it now has momentum (and kinetic energy) that carries it on past x = 0.

The block continues to x = – A, where its acceleration then becomes +kA/m.

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Analysis Model, Simple Harmonic Motion

What are the units of k/m, in ?

They are 1/s2, which we can regard as a frequency-squared, so let’s write it as

Then the equation becomes

A typical way to solve such a differential equation is to simply search for a function that satisfies the requirement, in this case, that its second derivative yields the negative of itself! The sine and cosine functions meet these requirements.

2

2

d x ka x

dt m

2 k

m

2a x

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SHM Graphical Representation

A solution to the differential equation is

A, are all constants:

A = amplitude (maximum position in either positive or negative x direction, = angular frequency,

= phase constant, or initial phase angle.A and are determined by initial conditions.

( ) cos( )x t A t

k

m

Remember, the period and frequency are:

2 1

2T f

T

Motion Equations for SHM

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22

2

( ) cos( )

( ) sin( )

( ) cos( )

x t A t

dxv t A t

dt

d xa t A t

dt

The velocity is 90o out of phase with the displacement and the acceleration is 180o out of phase with the displacement.

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SHM Example 1 Initial conditions at t = 0 are

x (0)= A v (0) = 0

This means = 0 The acceleration reaches extremes of 2A at A. The velocity reaches extremes of A at x = 0.

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SHM Example 2 Initial conditions at t = 0 are

x (0)= 0 v (0) = vi

This means = / 2 The graph is shifted one-quarter cycle to the right compared to the graph of x (0) = A.

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The spring force is a conservative force, so in a frictionless system the energy is constant

Kinetic energy, as usual, is

The spring potential energy, as usual, is

Then the total energy is just

Consider the Energy of SHM Oscillator

212 (a constant)E K U kA

2 2 21 12 2 cosU kx kA t

2 2 2 21 12 2 sinK mv m A t

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Transfer of Energy of SHM The total energy is contant at all times, and is

(proportional to the square of the amplitude) Energy is continuously being transferred between potential

energy stored in the spring, and the kinetic energy of the block.

212E kA

Simple Pendulum The forces acting on the bob are the

tension and the weight. T is the force exerted by the string mg is the gravitational force The tangential component of the

gravitational force is the restoring force.

Recall that the tangential acceleration is

This gives another differential equation

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2

2t

da r L L

dt

2

2sin (for small )

d g gm m

dt L L

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Frequency of Simple Pendulum

The equation for is the same form as for the spring, with solution

where now the angular frequency is

Summary: the period and frequency of a simple pendulum depend only on the length of the string and the acceleration due to gravity. The period is independent of mass.

max( ) cos( )t t

2 so the period is T = 2

g L

L g

Torsional Pendulum Assume a rigid object is suspended from a wire attached at its top to a fixed support. The twisted wire exerts a restoring torque on the object that is proportional to its angular position. The restoring torque is

is the torsion constant of the support wire.

Newton’s Second Law gives dI I

dt

d

dt I

2

2

2

2

Section 15.5