How Does Friction Affect Spring Oscillations?

[Tonyt88]

A spring with spring constant k stands vertically, and a mass m is placed on top of it. The mass is gradually lowered to its equilibrium position. With the spring held at this compression length, the system is rotated to a horizontal position. The left end of spring is attached to a wall, and the mass is placed on a table with coefficient of kinentic friction μ = 1/8. The mass is released.

(a) what is the initial compression of the spring?

(b) How much does the maximal stretch (or compression) of the spring decrease after each half-oscillation?

(c) How many times does the mass oscillate back and forth before coming to rest?

 

[Tonyt88]

Would the first part be F = ma = -kx
Thus: mg/k = x, or no?

 

[physics girl phd]

Looks like you have a good start (the system is in equilibrium with the force of the spring pushing up to balance the weight of the mass). Now what changes about the forces involved once you put the system on its side?

 

[acnewman]

less than helpful

 

[paranormal]

I would approach this problem by first defining the variables involved. The spring constant, k, represents the stiffness of the spring and is measured in units of force per unit length. The mass, m, is measured in units of mass. The coefficient of kinetic friction, μ, is a dimensionless quantity that represents the amount of resistance to motion between two surfaces in contact. With this information, we can proceed to answer the questions posed.

(a) To determine the initial compression of the spring, we can use the equation F = kx, where F is the force applied to the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position. In this case, the force applied is the weight of the mass, mg, and the displacement is the initial compression of the spring, denoted as x. Therefore, the initial compression of the spring is x = mg/k.

(b) The maximal stretch (or compression) of the spring will decrease after each half-oscillation due to the friction between the mass and the table. This friction will cause the amplitude of the oscillations to decrease over time. To determine the amount of decrease, we can use the equation Δx = μmg/k, where Δx is the change in the amplitude, μ is the coefficient of kinetic friction, m is the mass, and g is the acceleration due to gravity. Therefore, the maximal stretch (or compression) will decrease by Δx = (1/8)(mg)/k after each half-oscillation.

(c) The number of oscillations before the mass comes to rest can be determined by using the equation T = 2π√(m/k), where T is the period of the oscillation. The period is the time it takes for one complete oscillation. Therefore, the number of oscillations can be calculated by dividing the time the mass is in motion by the period of the oscillation. This will result in a whole number, as the mass will eventually come to rest due to the friction.

In conclusion, the initial compression of the spring can be calculated using the force and spring constant, the maximal stretch (or compression) will decrease after each half-oscillation due to friction, and the number of oscillations before the mass comes to rest can be determined using the period of the oscillation. These calculations help us understand the behavior of the system and how different factors, such as friction, can affect