oscillations & shm 03-08 lq

8/10/2019 Oscillations & SHM 03-08 LQ

1/5

93197825 A** [email protected]

Cambridge PhD, UC MA, HKU

1

1. Simple harmonic motion and the greenhouse effect(a) A body is displaced from equilibrium. State the twoconditions necessary for the body to

execute simple harmonic motion.1. .........................................................................................................................

.........................................................................................................................2. .........................................................................................................................

.........................................................................................................................(2)

(b) In a simple model of a methane molecule, a hydrogen atom and the carbon atom can beregarded as two masses attached by a spring. A hydrogen atom is much less massive thanthe carbon atom such that any displacement of the carbon atom may be ignored.The graph below shows the variation with time tof the displacementxfrom itsequilibrium position of a hydrogen atom in a molecule of methane.

The mass of hydrogen atom is 1.7 1027

kg. Use data from the graph above(i) to determine its amplitude of oscillation.

.........................................................................................................................(1)

(ii) to show that the frequency of its oscillation is 9.1 1013

Hz...................................................................................................................................................................................................................................................

(2)

(iii) to show that the maximum kinetic energy of the hydrogen atom is 6.2 1018J..........................................................................................................................

.........................................................................................................................

.........................................................................................................................(2)

(c) On the grid below, sketch a graph to show the variation with time tof the speed vof thehydrogen atom for one period of oscillation starting at t= 0. (There is no need to addvalues to the speed axis.)


2/5



2

(3)(d) Assuming that the motion of the hydrogen atom is simple harmonic, its frequency of

oscillationfis given by the expression

,2

1

pm

kf

=

where kis the force per unit displacement between a hydrogen atom and the carbon atomand mpis the mass of a proton.

(i) Show that the value of kis approximately 560 N m1

...................................................................................................................................................................................................................................................

(1)

(ii) Estimate, using your answer to (d)(i), the maximum acceleration of the hydrogenatom.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

(2)

(e) Methane is classified as a greenhouse gas.(i) Describe what is meant by a greenhouse gas.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................(2)

(ii) Electromagnetic radiation of frequency 9.1 1013Hz is in the infrared region ofthe electromagnetic spectrum. Suggest, based on the information given in (b)(ii),why methane is classified as a greenhouse gas.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

(2)

(Total 17 marks)

_____________________________________________________________________________


3/5



3

2. Simple harmonic motion and the greenhouse effect(a) A body is displaced from equilibrium. State the twoconditions necessary for the body to

execute simple harmonic motion.1. .........................................................................................................................

.........................................................................................................................2. .........................................................................................................................

.........................................................................................................................(2)

(b) In a simple model of a methane molecule, a hydrogen atom and the carbon atom can beregarded as two masses attached by a spring. A hydrogen atom is much less massive thanthe carbon atom such that any displacement of the carbon atom may be ignored.The graph below shows the variation with time tof the displacementxfrom itsequilibrium position of a hydrogen atom in a molecule of methane.

The mass of hydrogen atom is 1.7 1027

kg. Use data from the graph above(i) to determine its amplitude of oscillation.

.........................................................................................................................(1)

(ii) to show that the frequency of its oscillation is 9.1 1013

Hz...................................................................................................................................................................................................................................................

(2)

(iii) to show that the maximum kinetic energy of the hydrogen atom is 6.2 1018J..........................................................................................................................

.........................................................................................................................

.........................................................................................................................(2)

(c) Assuming that the motion of the hydrogen atom is simple harmonic, its frequency ofoscillationfis given by the expression

,2

1

pm

kf

=

where kis the force per unit displacement between a hydrogen atom and the carbon atomand mpis the mass of a proton.

(i) Show that the value of kis approximately 560 N m1...................................................................................................................................................................................................................................................

(1)

(ii) Estimate, using your answer to (c)(i), the maximum acceleration of the hydrogen


4/5



4

atom............................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................(2)

(d) Methane is classified as a greenhouse gas.(i) Describe what is meant by a greenhouse gas.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................(2)

(ii) Electromagnetic radiation of frequency 9.1 1013

Hz is in the infrared region ofthe electromagnetic spectrum. Suggest, based on the information given in (b)(ii),why methane is classified as a greenhouse gas.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

(2)

(Total 14 marks)

1. Simple harmonic motion and the greenhouse effect(a) the force acting / accelerating (on the body) is directed towards equilibrium

(position);and is proportional to its / the bodies displacement from equilibrium; 2

(b) (i) 1.5 1010

m; 1

(ii) T= 1.1 1012

s;

;101.1

112

=

f

= 9.1 1013

Hz 2

(iii) = (2f) = 5.7 1014(rad s1);

Emax= ) ( ) ( ) ;107.5105.1107.1 28220227212

0

2

21 = xm

= 6.2 1018J 2

(c) negative sine;starting at zero;

with same frequency as displacement; (allow 2mm square) 3

(d) (i) k= (42f

2mp) = 40 83 10

261.7 10

27;

560 Nm1 1

(ii) use of F= kxand F= ma;

to give ;ms100.5107.1

105.1560 21927

10

=

=a 2

(e) (i) infra red radiation radiated from Earth will be absorbed bygreenhouse gases;and so increase the temperature of the atmosphere / Earth; 2

(ii) the natural frequency of oscillation (of a methane molecule)

is equal to 9.1 1013

Hz;

because of resonance the molecule will readily absorb radiationof this frequency; 2

[17]


5/5



5

_____________________________________________________________________________2. Simple harmonic motion and the greenhouse effect

(a) the force acting / accelerating (on the body) is directed towardsequilibrium (position);

and is proportional to its / the bodies displacement from equilibrium; 2

(b) (i) 1.5 1010m; 1

(ii) T= 1.1 1012s;

;101.1

112

=

f

= 9.1 1013Hz 2

(iii) = (2f) = 5.7 1014(rad s1);

Emax= ( ) ( ) ( ) ;107.5105.1107.1 28220227212

0

2

21 = xm

= 6.2 1018J 2

(c) (i) k= (42f2mp) = 40 83 10261.7 1027;

560 Nm1 1

(ii) use of F= kxand F= ma;

to give ;ms100.5107.1

105.1560 21927

10

=

=a 2

(d) (i) infra red radiation radiated from Earth will be absorbed by

greenhouse gases;and so increase the temperature of the atmosphere / Earth; 2

(ii) the natural frequency of oscillation (of a methane molecule)

is equal to 9.1 1013Hz;

because of resonance the molecule will readily absorb radiationof this frequency; 2

[14]

oscillations & shm 03-08 lq

Documents