Solve JEE (MAIN) 2018 With A Smile – III

Let Us YOU Solve Physics JEE (MAIN) 2018 Physics Paper

In a series of blogs, we @ www.PhysicsAcademyOnline.com encourage our readers (students preparing for engineering and medical entrance exams) to solve on their own Physics questions from recent examination papers. We believe, and we want our readers to believe, a solid grip on the theories and quick problem-solving skill are the success keys to a high rank in a keenly contested exam. Presently, we are discussing JEE (Main) 2018, Paper-1, held on 08 April 2018.

Question 8: A particle is moving with a constant speed in a circular orbit of radius R under a central force inversely proportional to the n^th power of R. If the period of revolution of the particle is T, then:

(1) T∝ R ^3/2 for any n
(2) T∝ R^{n/2 + 1}
(3) T∝ R^{(n + 1)/2}
(4) T∝ R^n/2

Using the condition given in the question, express the central force, F_c , in terms of orbit radius, R. What is the acceleration of the particle? Discussed in detail here: Chapter Name – Mechanics; Category – Basic; Topic Name – Circular Motion; Video Name – Derivation of Centripetal Acceleration.

For an alternative expression, you may see: Chapter Name – Mechanics; Category – Basic;  Topic Name – Circular Motion; Video Name – Relations between Linear and Angular Quantities.

Use Newton’s second law to express F_c . Compare with its earlier expression. How do you introduce time period, T, in your last result? The relevant formula is here: Chapter Name – Mechanics; Category – Basic; Topic Name – Circular Motion; Video Name – Angular Quantities in Circular Motion.

Little rearrangement leads you to to the correct option.

Question 9: A solid sphere of radius r, made of a soft material of bulk modulus K, is surrounded by a liquid in a cylindrical container. A massless piston of area “a” floats on the surface of the liquid, covering the entire cross section of the container. When a mass m is placed on top of the piston to compress the liquid, the fractional decrement in the radius of the sphere, │dr/r│, is:

(1) Ka/mg           (2) Ka/3mg             (3) mg/3Ka            (4) mg/Ka

What is the formula for bulk modulus, K, of a material? The one using calculus notation is preferred. Watch the lecture: Chapter Name – Elasticity and Fluid Mechanics; Category –Basic;  Topic Name – Elasticity; Video Name – Bulk Modulus & Relations among Elastic Constants.
As the mass m is placed on the piston, what is the differential change in pressure? Therefore, what is the fractional change in volume of the sphere?

Now for a sphere, volume, V = 4 πr³/3. Taking logarithm on both sides, ln V = ln (4π/3) + 3 ln r. Differentiating both sides, dV/V = 3 dr/r. The fractional change in radius follows. By the way, how is “fractional decrement” different from “fractional change”?