Question

A massless spring of length L = 100 cm is placed on a frictionless table. Its left end is attached to a rigid support, and on its right end is attached a mass 1.2 kg. The spring constant is K = 200 N/m. The spring is pulled to the right by 15 cm, and then released, and it oscillates on the table. Calculate the a) frequency of oscillations (in hertz), b) maximum value of kinetic energy of the mass during its motion (in joules). c) maximum value of the speed of the mass during its motion (in m/s). Write the value of the speed in canvas.

Answer 1

The frequency of oscillations is approximately 3.62 Hz. The maximum value of kinetic energy is approximately 8.37 J. The maximum value of speed is approximately 4.08 m/s.

Step-by-step explanation:

To calculate the frequency of oscillations (in hertz), we can use the formula:

f = 1 / T

where T is the period of oscillation. The period can be found using the formula:

T = 2π√(m/K)

where m is the mass of the object and K is the spring constant. Plugging in the given values, we get:

T = 2π√(1.2 / 200) ≈ 0.276 s

Substituting this value into the frequency formula, we have:

f = 1 / 0.276 ≈ 3.62 Hz

The maximum value of kinetic energy can be calculated using the formula:

KE = (1/2)mv²

Given that the mass is 1.2 kg and the maximum displacement is 15 cm, we can find the maximum velocity using the formula:

v = √(2Kx/m)

Plugging in the given values, we have:

v = √(2 * 200 * 0.15 / 1.2) ≈ 4.08 m/s

Finally, we can substitute this value into the kinetic energy formula to find the maximum value:

KE = (1/2) * 1.2 * (4.08)² ≈ 8.37 J

The maximum value of the speed can be found by substituting the maximum velocity into the absolute value of the speed formula:

speed = |v|

Therefore, the maximum value of the speed is approximately 4.08 m/s.