Another simple harmonic motion

[pezola]

Homework Statement

A spring is hung from the ceiling. A 0.400 kg block is then attached to the free end of the spring. When released from rest, the block drops 0.150 m before momentarily coming to rest.

(a) What is the spring constant of the spring?

(b) Find the angular frequency of the block's vibrations.


Again, please explain what is going on here and the process to solve.

 

[lzkelley]

Consider the point at which the block comes to rest (after it has been attached and drops a little).
Can you draw a free body diagram?
What forces come into play?
How can you relate them to each other?

 

[Moetasim]

I can provide a response to the given content by explaining the concept of simple harmonic motion and how it applies to this scenario. Simple harmonic motion is a type of periodic motion in which a system oscillates back and forth around an equilibrium position with a constant amplitude and a constant period. In this scenario, the spring and the attached block are undergoing simple harmonic motion.

To solve for the spring constant, we can use the formula F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position. In this case, the force exerted by the spring is equal to the weight of the block, which is given by mg, where m is the mass of the block and g is the acceleration due to gravity. Since the block drops 0.150 m before coming to rest, x = 0.150 m. Therefore, we can set up the equation mg = -kx and solve for k, which gives us k = mg/x. Plugging in the values of m, g, and x, we get k = (0.400 kg)(9.8 m/s^2)/(0.150 m) = 26.13 N/m.

To find the angular frequency of the block's vibrations, we can use the formula ω = √(k/m), where ω is the angular frequency, k is the spring constant, and m is the mass of the block. Plugging in the values of k and m, we get ω = √(26.13 N/m)/(0.400 kg) = 7.24 rad/s. This means that the block's vibrations have an angular frequency of 7.24 radians per second.

In conclusion, the spring constant of the spring in this scenario is 26.13 N/m and the angular frequency of the block's vibrations is 7.24 rad/s. These values can be used to further analyze the simple harmonic motion of the spring and block system.