| Date | May 2014 | Marks available | 2 | Reference code | 14M.2.SL.TZ1.4 |
| Level | Standard level | Paper | Paper 2 | Time zone | Time zone 1 |
| Command term | Calculate, Deduce, and State | Question number | 4 | Adapted from | N/A |
Question
This question is in two parts. Part 1 is about solar radiation and the greenhouse effect. Part 2 is about a mass on a spring.
Part 1 Solar radiation and the greenhouse effect
The following data are available.
Part 2 A mass on a spring
An object is placed on a frictionless surface and attached to a light horizontal spring.
The other end of the spring is attached to a stationary point P. Air resistance is negligible. The equilibrium position is at O. The object is moved to position Y and released.
State the Stefan-Boltzmann law for a black body.
Deduce that the solar power incident per unit area at distance d from the Sun is given by
\[\frac{{\sigma {R^2}{T^4}}}{{{d^2}}}\]
Calculate, using the data given, the solar power incident per unit area at distance d from the Sun.
State two reasons why the solar power incident per unit area at a point on the surface of the Earth is likely to be different from your answer in (c).
The average power absorbed per unit area at the Earth’s surface is 240Wm–2. By treating the Earth’s surface as a black body, show that the average surface temperature of the Earth is approximately 250K.
Explain why the actual surface temperature of the Earth is greater than the value in (e).
Outline the conditions necessary for the object to execute simple harmonic motion.
The sketch graph below shows how the displacement of the object from point O varies with time over three time periods.
(i) Label with the letter A a point at which the magnitude of the acceleration of the object is a maximum.
(ii) Label with the letter V a point at which the speed of the object is a maximum.
(iii) Sketch on the same axes a graph of how the displacement varies with time if a small frictional force acts on the object.
Point P now begins to move from side to side with a small amplitude and at a variable driving frequency f. The frictional force is still small.
At each value of f, the object eventually reaches a constant amplitude A.
The graph shows the variation with f of A.
(i) With reference to resonance and resonant frequency, comment on the shape of the graph.
(ii) On the same axes, draw a graph to show the variation with f of A when the frictional force acting on the object is increased.
Markscheme
power/energy per second emitted proportional to surface area;
and proportional to fourth power of absolute temperature / temperature in K;
Accept equation with symbols defined.
solar power given by 4πR2σT 4 ;
spreads out over sphere of surface area 4πd 2 ;
Hence equation given.
\(\left( {\frac{{\sigma {R^2}{T^4}}}{{{d^2}}} = } \right)\frac{{5.7 \times {{10}^{ - 8}} \times {{\left[ {7.0 \times {{10}^8}} \right]}^2} \times {{\left[ {5.8 \times {{10}^3}} \right]}^4}}}{{{{\left[ {1.5 \times {{10}^{11}}} \right]}^2}}}\);
=1.4×103(Wm-2 );
Award [2] for a bald correct answer.
some energy reflected;
some energy absorbed/scattered by atmosphere; depends on latitude;
depends on time of day;
depends on time of year;
depends on weather (eg cloud cover) at location; power output of Sun varies;
Earth-Sun distance varies;
power radiated = power absorbed;
\(T = {}^4\sqrt {\frac{{240}}{{5.7 \times {{10}^{ - 8}}}}} = \left( {250{\rm{K}}} \right)\);
Accept answers given as 260 (K).
radiation from Sun is re-emitted from Earth at longer wavelengths; greenhouse gases in the atmosphere absorb some of this energy; and radiate some of it back to the surface of the Earth;
the force (of the spring on the object)/acceleration (of the object/point O) must be proportional to the displacement (from the equilibrium position/centre/point O);
and in the opposite direction to the displacement / always directed towards the equilibrium position/centre/point O;
(i) one A correctly shown;
(ii) one V correctly shown;
(iii) same period; (judge by eye)
amplitude decreasing with time;
(i) resonance is where driving frequency equals/is close to natural/resonant frequency;
the natural/resonant frequency is at/near the maximum amplitude of the graph;
(ii) lower amplitude everywhere on graph, bit still positive;
maximum in same place/moved slightly (that is, between the lines) to left on graph;
Examiners report
The Stefan-Boltzmann law was poorly understood with few candidates stating that the absolute temperature is raised to the fourth power.
This question was poorly done with few candidates substituting the surface area of the sun or the surface area of a sphere at the Earth’s radius of orbit.
Despite not being able to state or manipulate the Stefan-Boltzmann law most candidates could substitute values into the expression and calculate a result.
This question was well answered at higher level.
To show the given value there is the requirement for an explanation of why the incident power absorbed by the Earth’s surface is equal to the power radiated by the Earth, few candidates were successful in this aspect. Although most could substitute into the Stefan-Boltzmann equation they needed to either show that the fourth root was used or to find the temperature to more significant figures than the value given.
A surprising number of candidates could not explain the greenhouse effect. A common misunderstanding was that the Earth reflected radiation into the atmosphere and that the atmosphere reflected the radiation back to the Earth.
The conditions for simple harmonic motion were poorly outlined by most candidates. Few identified a relationship between force/acceleration and displacement, with most talking about it going backwards and forwards without slowing down.
This question was well answered by many. The only notable mistake was with reducing the time period of the damped oscillation.
i) Identifying the peak of the graph with the resonant frequency was broadly successfully done but not many candidates stated that this occurs when the driving frequency is equal to the natural frequency.
ii) This sketch was generally well done.
