A large, cylindrically-shaped weight is suspended from a long spring found hanging from a rafter. The weight is slender metal cylinder about two meters long that is suspended with its flat bottom about half meter above the level concrete floor. When a 1 kg mass is added, the large weight sank down an additional 3 mm. Then, after removing the mass, the larger weight is slowly made to oscillate up and down with a frequency of twelve oscillations per minute.

What is the force constant of the spring?
What is the mass of the unknown weight?
If the weight is pushed down 10cm and released, how fast is it moving at its maximum speed?

Use this statement to determine k:

<<When a 1 kg mass is added, the large weight sank down an additional 3 mm. >>

k = m*g/X
= (1 kg)*(9.8 m/s^2))/0.003 m
m is the added mass and X is the resulting extra deflection

Use that k value and this formula to obtain M:
w = 2*pi*f = sqrt(k/M)= 2*pi*(0.2 s^-1)

Note that 12 cycles per minute was converted to frequency in Hz.
w is the angular frequency

Use this formula to determine max speed, Vmax:

Vmax = w*(amplitude)

Thanks for your time

To find the force constant of the spring, 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.

When a 1 kg mass is added, the large weight sank down an additional 3 mm (0.003 meters). Since the weight is slender and cylindrical, we can assume it is vertically oriented, and the additional displacement will be equal to the elongation of the spring.

Therefore, we can calculate the force constant (k) using the formula:

F = k * x

Where F is the force applied, k is the force constant, and x is the displacement.

Given that a 1 kg mass is equivalent to 9.8 Newtons of force (acceleration due to gravity), and the additional displacement is 0.003 meters, we can substitute the values into the formula:

9.8 N = k * 0.003 m

Solving for k:

k = 9.8 N / 0.003 m
k ≈ 3266.67 N/m

So, the force constant of the spring is approximately 3266.67 N/m.

To find the mass of the unknown weight, we need to use the formula for the period of oscillation of a mass-spring system:

T = 2π * √(m/k)

Where T is the period, m is the mass, and k is the force constant.

We are given the frequency of oscillation, which is 12 oscillations per minute. To find the period in seconds, we can convert minutes to seconds and take the reciprocal of the frequency:

T = 1 / (12/60) = 5 seconds

Substituting the value of the period and the force constant into the formula, we get:

5 = 2π * √(m/3266.67)

Solving for m:

√(m/3266.67) = 5 / (2π)
m/3266.67 = (5 / (2π))^2

m = 3266.67 * [(5 / (2π))^2]

Calculating this expression will give you the mass of the unknown weight.

Now, let's calculate the speed when the weight is pushed down 10 cm (0.1 meters) and released, assuming the system is frictionless.

The maximum speed occurs when the weight reaches the lowest point of its oscillation, also known as the amplitude. To determine the speed, we can use conservation of mechanical energy, which states that the sum of kinetic energy and potential energy remains constant in a closed system.

At the extreme point of the oscillation, all potential energy is converted into kinetic energy. Therefore, we can equate the potential energy at the point of maximum displacement (P.E.) to the kinetic energy at the lowest point (K.E) and solve for velocity (v):

P.E. = K.E.
(1/2) k x^2 = (1/2) m v^2

Given that the displacement is 0.1 meters, the force constant (k) is known from the previous calculation, and the mass (m) can also be determined, we can substitute these values into the equation and solve for v:

(1/2) * 3266.67 N/m * (0.1 m)^2 = (1/2) * m * v^2

Solving for v will give you the speed of the weight at its maximum speed.