A spring is stretched 5 × 10−2 m by a force of 5 × 10−4 N. A mass of 0.01 kg is placed on the

lower end of the spring. After equilibrium has been reached, the upper end of the spring is
moved up and down so that the external force acting on the mass is given by F(t) = 20 cos wt.
Calculate (i) the position of the mass at any time, measured form the equilibrium position and
(ii) the angular frequency for which resonance occurs

Find here,

H,...www.assignmentexpert.com/homework-answers/physics/mechanics-relativity/question-85565....

To calculate the position of the mass at any time, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. The equation for Hooke's Law is:

F = -kx

Where F is the force applied to the spring, k is the spring constant, and x is the displacement from the equilibrium position.

Given that the spring is stretched by 5 × 10^(-2) m with a force of 5 × 10^(-4) N, we can rearrange Hooke's Law to solve for the spring constant:

k = -F / x

k = -(5 × 10^(-4) N) / (5 × 10^(-2) m)

k = - (5 × 10^(-4) N) / (5 × 10^(-2) m)

k = -10 N/m

Now, let's calculate the position of the mass at any time. Since the external force acting on the mass is given by F(t) = 20 cos(ωt), we can re-arrange Hooke's Law to solve for x:

x = -F / k

At any time t, the force acting on the mass is F(t) = 20 cos(ωt), so:

x(t) = - (20 cos(ωt)) / (-10 N/m)

x(t) = 2 cos(ωt)

This equation gives the position of the mass at any time t, measured from the equilibrium position.

Now, let's calculate the angular frequency for which resonance occurs. Resonance occurs when the driving frequency (angular frequency) matches the natural frequency of the system. The natural frequency of a mass-spring system is given by:

ω0 = sqrt(k / m)

Where ω0 is the angular frequency, k is the spring constant, and m is the mass.

Given that the mass is 0.01 kg and the spring constant is -10 N/m, we can calculate ω0:

ω0 = sqrt((-10 N/m) / (0.01 kg))

ω0 = sqrt(-1000 rad/s^2)

Since the natural frequency ω0 is imaginary (due to the negative square root), resonance does not occur in this system.

Therefore, the position of the mass at any time, measured from the equilibrium position, is given by the equation x(t) = 2 cos(ωt), and resonance does not occur.