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**Introductory Exercise 11.1**

**Ques 1: A particle moves under the force F(x) = (x ^{2} - 6x) N, where x is in metres. For small displacements from the origin what is the force constant in the simple harmonic motion approximation?**

F

= (x + Î”)

= 2xÎ”x - 6Î”x = -(6 - 2x) Î”x

= - 6Î”x for x = 0

kÎ”x â‡’ k = 6N/m

Sol:

= 10cm

x = A sin (Ï‰t + Î¸)

Sol:

= 15 cm = Amplitude.

Sol:

8 cm = 10 sin Ï‰t

= 1.4 Ã— 10

a) What is the total mechanical energy of the particle?

(b) What is the shortest time required for the particle to move from x = 0.5 m to x = - 0.75 m?

Sol:

**Ques 7: A body of mass 200 g is in equilibrium at x = 0 under the influence of a force F(x) = (-100x + 10x ^{2}) N(a) If the body is displaced a small distance from equilibrium, what is the period of its oscillations? (b) If the amplitude is 4.0 cm, by how much do we error in assuming that F(x) = - kx at the end points of the motion.Ans: **

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