Mechanics - Further Mathematics - Hooke's Law Questions (Elastic, Kinetic, and Potential Energy)

Topics Covered:
Law of Conservation of Energy (Elastic Potential Energy, Kinetic Energy, Potential Energy) - Hooke's Law
Circular Motion
Newton's Second Law of Motion
Questions Covered:
Question 1: A particle P of mass m is placed on a fixed smooth plane which is inclined at angle, theeta, to the horizontal. A light spring, of natural length a is and modulus of elasticity, 3 mg, has one end attached to P attached to P and the other end attached to fixed point O at the top of the plane. The spring lies along a line of the greatest slope of the plane. The system is released from rest with the spring at its natural length.
Find in terms of a and theeta, an expression for the greatest extension of the spring in the subsequent motion.
Question 2: One end of a light elastic string, of natura length a and modulus of elasticity 4mg, is attached to a fixed point O. The other end of a spring is attached to a particle of mass m. The particle moves in a horizontal circle with constant angular speed under root(g/a) with the string inclined at an angle theeta to the downward vertical through O. The length of the spring during this motion is (k + 1)a.
(a) Find the value of k.
(b)Find the value cos theeta.
Question 3:
One end of a light elastic string, of natural length a and modulus of elasticity k, is attached to a particle P of mass m. The other end of the string is attached to a fixed point Q. The particle P is projected vertically upwards from Q. When P is moving upwards and at a distance 4a/3 directly above Q, it has a speed under root(2ga). At this point, its acceleration is 7g/3 downwards.
Show that k = 4mg and find in terms of a the greatest height above Q reached by P.
Mathematics, A-Level, Mechanics, Hooks Law, A-Level Mathematics, Applied Mathematics,
Mathematics(9709), M1 of Further MATHEMATICS (9231/03),
Engineering Mathematics, Mechanics

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