BE FAST-NUCES — National University of Computer & Emerging Sciences

NOTE: Attempt FIVE questions, selecting not more than TWO questions from each section.

Section-I

Q.1:

(a)-What is spherical polar coordinate system? Derive the relation between

spherical polar coordinate system and three dimensional Cartesian coordinate system.

Also express three spherical polar coordinate unit vectors

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in terms off, / and k.

(b)-A particle moves along the curve x = 2t?, y = t? - 4t, ≥ = 3t - 5, where t is time.

Find components of its velocity and acceleration at t = 1 in the direction ! -

3) + 2k.

(c)-Does the value of vector quantity depend upon selected coordinate system?

Q.2:

(a)-Derive formula for relativistic variation of mass with velocity. Hence prove that

it is not possible for material particle to have a velocity equal to or greater than the

velocity of light.

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(b)-If kinetic energy K and momentum p of a particle can be measured, it should be

possible to find its mass and thus identify the particles. Show that

2Kc*

(e)-The sweep rate of the tail of a comet can exceed the speed of light. Explain this

phenomenon and show that there is no contradiction with relativity.

Q.3:

(a) - State Newton's law of universal gravitation and write its vector form. How

gravitational constant was determined by Cavendish.

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(b)-An artificial satellite is in a circular orbit about earth. State how varies period,

kinetic energy, angular momentum and speed with r.

(e)-The gravitational force exerted by the sun on the moon is about twice as great as

the gravitational force exerted by the earth on the moon. Why then does the moon not

escape from the earth?

Q.4:

(a)-State parallel axis theorem. Using parallel axis theorem, find rotational inertia

of a solid rod of length L and mass M, about an axis which is perpendicular to its

length and passes through one of its end.

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(b)-A solid cylinder of mass M and radius R rolls without slipping down an inclined

plane of length L and height h. Prove that speed of its center of mass when the

cylinder reaches bottom is,

4 gh

Vem =

(c)-If we have two spheres of equal masses of copper and lead, then which sphere will

be rotating fast?

Section-II

Q.5:

(a)-Prove that simple harmonic motion can be described as the projection of

uniform circular motion along the diameter of a circle. Find equations of displacement,

velocity and acceleration of projection. Hence show that the circular motion can be

regarded as combination of identical simple harmonic motion but different in phase

90°

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J-If a pendulum has a period of 1s at the equator. What would be its period at the

south Pole?