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## The Handbook of Mathematics, Physics and

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KEELE UNIVERSITY

EXAMINATIONS, 2010/11

Level I

Tuesday 24 th

May 2011, 09.30-11.30

PHYSICS/ASTROPHYSICS

PHY-10020

OSCILLATIONS AND WAVES

Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered on the exam paper; PART C should be answered in the examination booklet which should be attached to the exam paper at the end of the exam with a treasury tag.

PART A yields 16% of the marks, PART B yields 24%, PART

C yields 60%.

Please do not write in the box below

A

B

C1

C2

C3

C4

Total

NOT TO BE REMOVED FROM THE EXAMINATION HALL

PHY-10020 Page 1 of 12

PART A Tick one box by the answer you judge to be correct

(marks are not deducted for incorrect answers)

A1 A block of mass m, hanging from a spring with force constant k and natural length L, oscillates purely in the vertical direction. The angular frequency of the oscillation is

ω = q k/m

ω = q mg/k

ω =

ω = q q g/L k/mg [1] where g is the acceleration due to gravity.

A2 A simple pendulum on the surface of the Earth has a period of 1 second. On the moon, where the acceleration due to gravity is 6 times lower, the period of the same pendulum would be

6 seconds

√

6 seconds

1

6

√

6 seconds seconds [1]

A3 An object is in simple harmonic motion about an equilibrium position, with an angular frequency of 3 s −1 and an amplitude of 0.4 m. The speed of the object at the equilibrium position is

3.6 m s −1 2.4 m s −1 1.2 m s −1 0 [1]

A4 A particle is in simple harmonic motion. At how many times during one oscillation cycle are the kinetic and potential energies of the particle equal to each other?

eight times four times two times one time [1]

A5 An oscillator with mass 300 g and natural angular frequency 6.00 s −1 is damped by a force F damp

= −γ ˙x . The critical damping constant is

γ = 1.80 kg s

−1

γ = 3.60 kg s

−1

γ = 2.55 kg s

γ = 10.8 kg s

−1

−1

[1]

/Cont’d

PHY-10020 Page 2 of 12

A6 The amplitude of a particular underdamped oscillator decays as

A(t) ∝ e −4 t . The total mechanical energy of the oscillator depends on time as

E tot

(t) ∝ e

−2 t

E tot

(t) ∝ e −8 t

E tot

(t) ∝ e

−4 t

E tot

(t) ∝ e −16 t

2

[1]

A7 The steady-state displacement and velocity of a forced harmonic oscillator are in phase with the external force

90 ◦ out of phase with the external force in phase with each other

90 ◦ out of phase with each other [1]

A8 A damped oscillator with natural angular frequency ω

0 is driven by an external force with angular frequency ω e

. The oscillator is shifted to a new equilibrium position if

ω

ω e e

= 0

= ω

0

ω e

=

√

2 ω

0

ω e

≫ ω

0

[1]

A9 In the scheme of analogies between electrical circuits and mechanical oscillators, the current in a circuit corresponds to the mass displacement velocity kinetic energy [1] of a mechanical system.

A10 The wave described by y(x, t) = e x−3t

(2x − 6t)

2

, with x and y in metres and t in seconds, propagates with a velocity of

3 m s −1 to the right

3 m s −1 to the left

6 m s

6 m s

−1

−1 to the right to the left [1]

/Cont’d

PHY-10020 Page 3 of 12

A11 The wavenumber k of a harmonic wave is, in standard notation, k = mω

2 k = ωv k = 2π/λ k = 2πλ [1]

A12 In the wave y(x, t) = 0.03 sin(4x + 8t) (where x and y are in metres and t is in seconds), the particle velocity at x = 0 at t = 0 is

2 m s −1

0.12 m s −1

0.24 m s

0

−1

[1]

A13 Two travelling harmonic waves combine to produce the standing wave y(x, t) = 0.01 sin(40x) cos(60t) (for x and y in metres, and t in seconds). The amplitude of each of the travelling waves is

A = 0.005 m

A = 0.02 m

A = 0.01 m

A = 0.1 m [1]

A14 A string of length L with both ends fixed vibrates in its n th harmonic.

The distance between adjacent nodes on the string is

2L/n L/n nL 2nL [1]

A15 Two waves with the same intensity I

0 interfere at a point P in space.

The maximum possible intensity of the total wave at P is

I

0

/ 2 I

0

2 I

0

4 I

0

[1]

A16 Interference patterns of the type seen in Young’s double-slit experiment arise when waves emitted in phase by two sources arrive at a point in space from opposite directions having travelled different distances at different times with slightly different frequencies [1]

/Cont’d

PHY-10020 Page 4 of 12

PART B Answer all EIGHT questions

B1 A block of mass m = 100 g attached to a horizontal spring with k = 40 N m

−1 has a displacement given by x(t) = 0.05 sin(ωt) m .

Calculate the velocity at time t = T /2, where T is the period of oscillation.

[3]

B2 An object of mass m = 0.4 kg is in simple harmonic motion about x = 0 with angular frequency ω = 3 s

−1

. Its total mechanical energy is E tot

= 4.5 × 10 −3 J. Find the speed of the object when its displacement is x = 0.03 m.

[3]

PHY-10020 Page 5 of 12

/Cont’d

B3 A particular damped harmonic oscillator has the equation of motion

For what value(s) of the constant γ in this equation will the motion be overdamped?

[3]

B4 A particular forced harmonic oscillator has the equation of motion x + 0.16 ˙x + 0.64 x = 1.44 cos(ω e t) .

Determine the value of ω e that gives velocity resonance. Sketch the steady-state velocity amplitude as a function of ω e in general.

(You are not required to calculate any numerical values of the amplitude.) [3]

PHY-10020 Page 6 of 12

/Cont’d

B5 Give a sketch illustrating the two normal modes of oscillation for a coupled pair of identical blocks on identical springs.

[3]

B6 A long string carries a transverse harmonic wave travelling in the negative-x direction with amplitude 2 cm, wavelength 60 cm, and frequency 440 Hz. The displacement of the string at x = 0 at t = 0 is y = 0. Write the wave function, y(x, t).

[3]

PHY-10020 Page 7 of 12

/Cont’d

B7 A travelling wave has the function y(x, t) = 8x

2

+ 6x + 8xt + 3t + 2t

2 for x in centimetres and t in seconds. Use the one-dimensional wave equation to find the phase speed of the wave.

[3]

B8 A transverse wave travels at speed 330 m s −1 on a piano wire that has a total mass of 10 grams and a length of 64 cm. What is the tension in the wire?

[3]

PHY-10020 Page 8 of 12

/Cont’d

PART C Answer TWO out of FOUR questions

C1 (a) A block of mass m on the end of a horizontal spring with force constant k is in simple harmonic motion about x = 0, with amplitude A.

i. Use the work-energy theorem to show that the potential energy of the block is U =

1

2 kx

2

.

[6] ii. Sketch the potential and kinetic energies of the block as functions of the displacement x from equilibrium.

[6]

(b) The potential energy of a simple pendulum with bob mass m and length L is

U (s) = mgL [1 − cos (s/L) ] where s is the arc length from the bottom of the swing, and g is the acceleration due to gravity.

i. Find the values of s for which U is either a minimum or a maximum. What is the net force on the bob at each of these positions? Which of the positions is a stable equilibrium?

[12] ii. Infer formulae for the effective spring constant, and the angular frequency of small-amplitude oscillations, of a simple pendulum.

[6]

PHY-10020 Page 9 of 12

/Cont’d

C2 (a) The displacement of an undriven, underdamped harmonic oscillator is given by x(t) = A

0 e −γ t/(2m) sin (ωt + φ

0

) , in which

ω ≡ q

ω

2

0

− γ

2

/(4m

2

) .

i. Sketch a representative x(t) curve, indicating clearly all main physical features of the motion.

[6] ii. A block with m = 0.4 kg is attached to a damped spring having k = 2.5 N m

−1 and γ = 0.56 kg s

−1

. The block is in equilibrium at t = 0, when it receives an impulse giving it an initial velocity of +0.6 m s −1 .

A. Verify that this system is underdamped.

[4]

B. Determine the displacement and the velocity of the block as functions of time for t > 0.

[14]

(b) Explain what is meant by the transient and the steady state for the motion of an underdamped oscillator that is driven by an external force of the form F (t) = F

0 cos(ω e t). Write down the general form of the displacement x(t) in the steady state.

[6]

PHY-10020 Page 10 of 12

/Cont’d

C3 (a) Give an argument as to why a wave travelling with speed v in one dimension must depend on position x and time t only in one of the combinations (x − vt) or (x + vt).

[6]

(b) The function y(x, t) = A sin [k(x − vt) + φ

0

] describes a travelling harmonic wave. For such a wave: i. The wavelength λ is defined as the smallest length such that y(x + λ, t

0

) = y(x, t

0

) for any x at a fixed t

0

. Use this to derive the standard relation between k and λ.

[6] ii. Show that y undergoes simple harmonic oscillation at any fixed position x in the wave. Thus, express the angular frequency ω of the wave in terms of k and v.

[6]

(c) Consider the function y(x, t) = 4 e x−2t

− e

3x−6t

.

i. Verify that this function is a solution to the one-dimensional wave equation.

[6] ii. Show that

∂

2 y/∂t

2

+ 8 ∂y/∂t + 12 y = 0 .

Thus, what kind of oscillation drives this wave?

[6]

PHY-10020 Page 11 of 12

/Cont’d

C4 (a) The displacement of a string vibrating in its third harmonic with both ends fixed is y(x, t) = 0.02 sin (0.2π x) cos (25π t) , where x and y are in centimetres and t is in seconds.

i. Calculate the wavelength of this standing wave, and the length of the string.

[4] ii. Find the positions of all nodes on the string, and sketch the wave at t = 0.

[6]

(b) Two sources, S

1 and S

2

, emit harmonic waves in phase with the same amplitude, frequency, and wavelength. These waves interfere at a point P , which is a distance x

1 from source S

1 and a distance x

2 from source S

2

. Show that the total wave at P is y tot

(P ) = 2A cos

k(x

1

− x

2

)

2

sin

k(x

1

+ x

2

2

)

− ωt + φ

0

. [6]

(c) Two identical speakers placed at x = 0 and x = 60 m emit sound in phase, with wavelength λ = 2 m. Use the equation for y tot

(P ) from part (b) to answer the following: i. Find all positions x along the straight line between the speakers, at which the total volume is a maximum.

[8] ii. Calculate the intensity of the sound at x = 30.25 m, relative to the maximum possible intensity.

[6]

PHY-10020 Page 12 of 12

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