Physics GS TIFR- Tata Institute of Fundamental Research Questions 11-13_ Solved

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11. I have two identical glass beakers, X and Y, full of water. I introduce a concentrated drop of ink into each beaker. I stir beaker Y until the ink concentration is uniform. What can you say about the diffusive motion of the ink molecule?

(i) The molecule diffuses with the same diffusion coefficient in both X and Y.

(ii) The molecule diffuses in X but not in Y.

(iii) The molecule diffuses in Y but not in X.

Answer: The diffusive motion of the ink molecule in Beaker Y is more significant than that in Beaker X. Stirring Beaker Y has increased the rate of diffusion of the ink molecules by mixing them with the water, causing the ink concentration to become uniform throughout the beaker. In contrast, Beaker X remains undisturbed, resulting in slower diffusion of the ink molecules, and a concentration gradient exists in the beaker, with the highest concentration near the point of introduction of the ink drop. The process of stirring Beaker Y has increased the molecular diffusion rate, leading to more rapid mixing of the ink and water molecules.

(iv) The molecule diffuses at the site of the original droplet but not close to the boundary.

Answer: The statement suggests that the molecule is undergoing diffusion, which is the process by which particles move from an area of higher concentration to an area of lower concentration.

If the molecule is diffusing at the site of the original droplet but not close to the boundary, this could be due to several factors. One possibility is that the concentration of the molecule is higher at the site of the original droplet, causing it to diffuse more readily in that area. Another possibility is that there are barriers or obstacles near the boundary that are impeding the diffusion of the molecule in that region.

Diffusion can be affected by a number of factors, including the size and shape of the diffusing particle, the temperature, the pressure, and the nature of the surrounding medium. Understanding the factors that affect diffusion can be important in a range of applications, from drug delivery to environmental monitoring.

12. Elon Musk goes cycling on the moon, where there is no air resistance. He finds a hill which has a nice smooth slope on one side, and a cliff on the other. If he jumps off the cliff his speed is greater that when he rides to the bottom of the slope, even though the starting height, and therefore potential energy is the same. His cycle has no friction. What is the explanation?

(i) The conversion of potential energy is more efficient if the resulting kinetic energy is along the same vector as the force due to gravity.

(ii) The distance traveled down the slope is greater, so it takes longer, and the final speed is slower.

(iii) The gas in his space suit applies friction even though the cycle has none.

(iv) The wheels of the bicycle acquire kinetic energy when he rides down the cliff.

Answer: The reason that Elon Musk would have a higher speed after jumping off the cliff than riding down the slope is due to the conservation of energy. Although his starting potential energy is the same in both cases, the jump off the cliff allows for a conversion of potential energy to kinetic energy without any loss due to air resistance or friction.

When Musk rides down the slope, some of the potential energy is converted into kinetic energy to overcome friction and air resistance, so he does not reach the same speed as he would from the jump. However, when he jumps off the cliff, there is no friction or air resistance to slow him down, so all of his potential energy is converted into kinetic energy, resulting in a higher speed.

Additionally, the slope may not be steep enough to allow Musk to accelerate to a high speed, whereas the jump off the cliff would provide a steeper initial acceleration due to the force of gravity acting over a shorter distance.

Overall, the lack of air resistance and friction on the moon allows for more efficient conversion of potential energy to kinetic energy, resulting in higher speeds in situations like jumping off a cliff.

13. The Pauli spin matrices are defined as

Which of these matrices has an eigenvector [1 -1]T?

(i) σx

(ii) σy

(iii) σz

(iv) None have real eigenvectors.

Answer: The Pauli spin matrices are:

σ1 = | 0 1 | σ2 = | 0 -i |

| 1 0 | | i 0 |

σ3 = | 1 0 |

| 0 -1 |

To determine which of these matrices has an eigenvector [1 -1]T, we can simply multiply each of the matrices by [1 -1]T and see which one yields a scalar multiple of the same vector.

Let's start with σ1:

σ1 [1 -1]T = | 0 1 | | 1 | = | 1 |

| 1 0 | |-1 | |-1 |

So σ1 has [1 -1]T as an eigenvector with eigenvalue 1.

Next, let's try σ2:

σ2 [1 -1]T = | 0 -i | | 1 | = |-i |

| i 0 | |-1 | | i |

So σ2 does not have [1 -1]T as an eigenvector.

Finally, let's try σ3:

σ3 [1 -1]T = | 1 0 | | 1 | = | 1 |

| 0 -1 | |-1 | |-1 |

So σ3 also has [1 -1]T as an eigenvector with eigenvalue -1.

Therefore, the matrix that has [1 -1]T as an eigenvector is σ1 and σ3.

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