The process of plucking of a guitar string results in the frequency of vibration being a function of the young’s modulus, the length and density of the string. Use dimensional analysis to derive an expression for the frequency of vibration of the string.
The wave in a string is a transverse wave which depends on the tension and the mass per unit length. If you double the tension the mass per unit length goes down by a small amount (the string gets a bit "thinner" because it gets longer) . Both these things increase the velocity of the transverse wave which
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A profit maximizing monopolist produces a quantity of a commodity. His
D(p) = 30 , where p is the price. Find the values of p, q and the profit.
A particle describe a central orbit ra = aa cos n under a force directed towards the pole. Find the law of Force.
(Rewrite the equation in terms of , take logarithms and differentiate with respect to
. Calculate
and use 3a) to find F.)
Suppose we find the sum of solutions obtained in (i) and (ii) of (b) above. Does this give a solution of the equation describing the motion? If yes, what physical situation, this solution will be correspond to? When is the velocity of the pendulum maximum? What type of energy does the pendulum have at position of maximum velocity?
For the problem of finding the period of oscillation of a simple pendulum, obtain a formulation in the case when air resistance is proportional to the square of the velocity. Find a solution for this formulation.
A particle is projected from the base of an inclined plane making an angle a with the horizontal. The initial velocity of projection is making an angle with the horizontal.
i) Derive the equation to the trajectory.
ii) Find the point at which the particle strikes the plane.
The process of plucking of a guitar string results in the frequency of vibration being a function of the young’s modulus, the length and density of the string. Use dimensional analysis to derive an expression for the frequency of vibration of the string.
Let a, K and 0 N be positive real numbers such that 0 < N0 <K. Find the solution of the logistic equation
If N0, N1, and N2 be the values of N at t = 0, t = T and t = 2T respectively, then show that
Also if at time t = t , the population N reaches half its carrying capacity then what is the form of the solution of the logistic equation?