Question

A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 6.60 N is applied. A 0.400-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x = 5.00 cm and released from rest at t = 0. (Assume that the direction of the initial displacement is positive. Use the exact values you enter to make later calculations.) (a) What is the force constant of the spring? N/m (b) What are the angular frequency ?, the frequency, and the period of the motion? ? = rad/s f = Hz T = s (c) What is the total energy of the system? J (d) What is the amplitude of the motion? cm (e) What are the maximum velocity and the maximum acceleration of the particle? vmax = m/s amax = m/s2 (f) Determine the displacement x of the particle from the equilibrium position at t = 0.500 s. cm (g) Determine the velocity and acceleration of the particle when t = 0.500 s. (Indicate the direction with the sign of your answer.) v = m/s a = m/s2

Answer 1

SOLUTION

Part (a)

The constant force can be obtained using:

Where: is the force applied and is the displacement. Isolating from equation (1) we have:

Replacing values we get:

Solving we obtain:

Part (b)

The angular frequency is defined as:

Where: is the constant force and is the mass of the object.

Replacing (2.1) and data given values into equation (3) we have:

Solving we obtain:

Now, the frequency can be obtained using:

Isolating from equation (4) we have:

Inserting (3.1) into equation (5) we have:

Solving we obtain:

Similarly, period is defined as:

Replacing (6) into equation (7) we get:

Solving we obtain:

Part (c)

The total energy of the system is defined as:

Where, is the amplitude.

Replacing (2.1) and data given values into equation (9) we get:

Solving we obtain:

Part (d)

On this case the amplitude is the initial displacement:

Part (e)

The maximum velocity is defined as:

Replacing (3.1) and (11) into equation (12) we have:

Solving we obtain:

Similarly, the maximum acceleration is defined as:

replacing (3.1) and (11) into (14) we have:

Solving we obtain:

Part (f)

The displacement of the particle can be obtained using:

Where: is the time and the constant phase , and this case .

Replacing values into equation (16) we get:

Solving we obtain:

Part (g)

The velocity is defined as:

Replacing data given values into (18):

Solving we obtain:

Similarly, the acceleration is defined as:

Repalcing values:

Solving we obtain: