Oxford PAT 2009, Questions 13 – 17

2009_page6

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7 thoughts on “Oxford PAT 2009, Questions 13 – 17

    1. Imagine the boat weighs nothing.

      When the anchor is in the boat, because the boat is floating, it displaces a volume of water V1; by Archimedes’ principle V1 is the volume of water whose weight is the weight of the anchor, so we can say V1 = Volume(anchor) x Density(anchor) / Density(water).

      When the anchor is on the bottom of the lake, the boat doesn’t displace any water because it weighs nothing, but the anchor lying on the bottom of the lake displaces a volume of water V2; V2 is the volume of water which is equal to the volume of the anchor, so we can say V2 = Volume(anchor).

      Looking at our two equations, we can see that V1 > V2 because Density(anchor) > Density(water). (We know this must be true because otherwise the anchor would float, which would mean it wasn’t really an anchor).

      The more water you displace, the higher the water level in the lake. Therefore because we displace less water when the anchor is on the bottom, the water level will go down.

    1. Hi Harry, thanks for your question.

      The answer (d) is correct but I guess you have correctly spotted that the explanatory notes that I wrote beside the answer are just totally wrong. I think I must have misread the question as saying that the angle subtended by Titan viewed from Saturn is 9.3mrad and the angle subtended by the Sun viewed from Earth is 4.4mrad, in which case the reasoning as written down would have been correct.

      But actually the question says that Titan viewed from Saturn subtends an angle of 4.4mrad, and the Sun viewed from Earth subtends an angle of 9.3mrad. So if we can work out what angle the Sun subtends when viewed from Saturn, then we will know the angles that both Titan and the Sun subtend when viewed from Saturn and we will be able to answer the question.

      To find the angle subtended by the Sun you need to know that:
      1. The angle subtended by an object is inversely proportional to its distance. This should be pretty obvious if you draw a diagram showing the object and the (small) angle subtended by it. When the angle is small, the size of the object is roughly the same as the length of arc that it covers, and the angle subtended is defined to be the length of arc divided by the distance to the object.
      2. Saturn is much further from the Sun than the Earth is. In fact it is over nine times further away.

      From (1) and (2) you can conclude that viewed from Saturn the Sun subtends an angle of roughly 9.3 x 1/9, which is about 1mrad. So this means that Titan looks much bigger from Saturn than the Sun does. Notice that you don’t need to know that Saturn is nine times as far away from the Sun as the Earth is (this would be a bit too obscure), but you should to know enough about the solar system know that it is a fair bit more than three times as far away, and therefore the Sun looks smaller than Titan when viewed from Saturn.

      Since the Sun looks smaller than Titan, we can see that it is impossible to have an annular solar eclipse on Saturn due to Titan. An annular eclipse only happens when the Sun is obscured by an object that subtends a smaller angle than it, leaving an unobscured ring (or annulus, from the Latin word meaning ‘little ring’).

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