A particle undergoes two simple harmonic motions, one along the X-axis and the other along a line at an angle of 45^ to the X-axis. These motions are described by x = x₀ t and s = s₀ t . Determine the amplitude of the resultant motion.

Step-by-step solution

The two simple harmonic motions are given by x = x₀ t , i and s = s₀ t ( 45^ i + 45^ j ) . Since both simple harmonic motions are in the same phase, the resultant amplitude is the vector sum of their individual amplitudes. The angle between the two amplitude vectors is = 45^ . Using the vector addition formula, the resultant amplitude A is given by: A = x₀^2 + s₀^2 + 2 x₀ s₀ 45^ A = x₀^2 + s₀^2 + 2 x₀ s₀ ( 1 2 ) A = [ x₀^2 + s₀^2 + 2 x₀ s₀ ]^ 1/2