Mass and Spring

Syntax

Overview

The most common example of simple harmonic motion is a mass on a spring. The force of the spring follows Hooke's Law. When a mass and spring are at rest they are considered at equilibrium because the force upward is equal to the force of gravity. Oh but the spring does have a restoring force, which is when and object is displaced from its equilibrium position and is then pushed back.

Equations

Spring constant (k) = -(F / d)
alpha = -kmx
Period (T) = 2*pi (sqrt(m / k)
Angular frequency (w) = sqrt(k/m)
f = (1/2pi)(sqrt(k/m))
vertical spring motion: kx - mg = ma - -> accelerartion = 0 - -> kx₀ = mg
F_net = k(x-x₀) = ma
U_spring = (1/2)(kx²)
E_tot = K + U_spring = (1/2)mv² + (1/2)kx² = (1/2)kA²

Question 1

There are two blocks (m=1.0 kg and M = 10 kg) and a spring (k = 200 N/m) are arranged on a horizontal, frictionless surface. The coefficientof static friction between the two blocks is 040. What amplitude of simple harmonic motion of the spring-blocks system puts the smaller block on the verge of slipping over the larger block?

Answer

w = 2pi*f
f = (mu)(N)
w = sqrt(k/m)
w = sqrt(200/10+1.0) = sqrt(200/11) = 2pi(0.40)N
N = 1.697

Question 2

A block weighing 14.0N, which slides without friction on a 40.0 degrees incline, is connected to the top of the incline by a massless spring of unstretched length 0.450 m and spring constant 120 N/m. (a) How far from the top of the incline does the block stop? (b) If the block is pulled slightly down the incline and released, what is the period of the resulting oscillations.

Answer

mgsin(theta) = kx
x = (mgsin(theta))/k
x = (((14.0)(10)sin(40))/120)-.450
x = 0.419

Special Cases

A special case for a mass on a spring would be a rotating mass on a spring, but this case is not likely to occur on the AP Physics exam, so I don't feel the need to display examples of this.

Applications

There aren't any real life applications unless you consider things dealing with Simple Harmonic Motion, which is a subject that I'm not covering.

External Links

• [http://www.myteacherpages.com/webpages/Lalewine/files/oscillation%20and%20gravitation%20keyconcepts.pdf] - Further Explanation
• [http://www2.krellinst.org/AiS/textbook/unit2/spring.html] - More Information
• [http://www.youtube.com/user/MindofMatthew] - This is really good physics entertainment, so you should watch this!!!

Joke