ch15 shm

.Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics, Copyright 2005 by Wiley and Sons

Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to Copyright Protection. 1

Ch15-1/18

Ch 15: Oscillations: Simple harmonic motion sections 1-4

Snapshots at equal time intervals. Vectors show speed which is max at x=0, minimum at x = +-xm

If we define t=0 when x = xm then it returns at t=T to x=xm, i.e. the period or time to complete one cycle is T.

The frequency, f, is the number of complete oscillations per second.

T = 1/f

SI unit of f is hertz, Hz = s-1

Ch15-2/18

periodic/harmonic motion

Any motion which repeats itself at regular intervals.

We are interested in a particular form:

x(t): the displacement at time t (m)

xm: the amplitude (m) (positive, maximum displacement)

ω: the angular frequency (s-1)

t: time (second)φ : phase constant or phase angle (radian)ωt + φ: phase

This is simple harmonic motion.



Ch15-3/18

Angular frequency

We know x(t) must return to the same value after T (and simplifying by taking φ = 0):

The cosine function repeats when 2π is added to argument:

SI unit of angular frequency is is the radian per second.

φ is in radians

Ch15-4/18

Examplesxm’ (red) > xm

T’ (red) = T/2

ω’ = 2 ω

φ = -π/4 for red

This is effect of a phase difference. There is a phase difference of 2π for 1 period T



Ch15-5/18

Velocity of SHM

vm is velocity amplitude

Ch15-6/18

Acceleration of SHM

am is acceleration amplitude

In SHM, acceleration is proportional to displacement but opposite in sign and related by ω2



Ch15-7/18

The force law for SHM

SHM is executed by a particle subject to a restoring force proportional to displacement

of particle.

From Newton’s second law, knowing acceleration => force known:

This is a restoring force proportional to the displacement, i.e., Hooke’s law for a spring

Ch15-8/18

Linear simple harmonic oscillator

Block-spring system: block mass m, frictionless surface. No force when x = 0.

Once pushed or pulled away from x=0, it moves in SHM.

It is a linear simple harmonic oscillator (linear oscillator). Linear since F proportional to x, not x2 etc. Using gives the angular frequency of

motion as: and period as:



Ch15-9/18

Question

Which of the following Force relationships implies simple harmonic oscillation?

(a) F = -5 x

(b) F = -400x2

(c) F = 10x

(d) F = 3x2

Ch15-10/18

Question

Which of the following Force relationships implies simple harmonic oscillation?

(a) F = -5 x

(b) F = -400x2

(c) F = 10x

(d) F = 3x2

Ans: (a) since F is proportional to the displacement AND is restorative.

To identify SHM, either

a = -(some +ve constant) x

or F= -(some +ve constant) x



Ch15-11/18

Question

Which of a, b or c is equivalent to ?

Ch15-12/18

Question

Which of a, b or c is equivalent to ?

Ans: (a) just because it is, by definition



Ch15-13/18

Sample problem 15-2For a block-spring system, given x(0) = -8.50 cm, v(0) = -0.920 m/s and a(0) = +47.0 m/s2. Determine angular frequency, phase constant, amplitude.

Key ideas: Write down:

from which we get:

Also

Ch15-14/18

Sample problem 15-2

There are two solutions:

which will lead to two values of xm:

Which is correct?



Ch15-15/18

Sample problem 15-2

There are two solutions:

which will lead to two values of xm:

Which is correct?Ans: The 155o solution since the amplitude must be positive.

Ch15-16/18

Energy in simple harmonic motion

The potential energy of a linear oscillator is associated with the spring – how much it is compressed or extended. From work with Hooke’s law we know

so that here we have

The kinetic energy is associated entirely with the block (we assume a massless spring) so that:

substitute using

and hence:



Ch15-17/18

Energy in simple harmonic motion (cont)

So total mechanical energy in system is:

Oscillating systems store potential energy in their springiness and kinetic energy in their inertia.

Ch15-18/18

Energy in simple harmonic motion (cont)

E as a function of time:

Note both U and K peak twice per period T (sin2, cos2)

E as a function of displacement:

ch15 shm

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