Question

Simple Harmonic Oscillations As it turns out, in an ideal system with no friction or air resistance, the object will go on oscillating up and down forever. This kind of motion is called simple harmonic motion or simple harmonic oscillations. The periodic trajectory of the oscillating mass over time is shown in the figure above. The amplitude of the oscillations, A, describes how far away from equilibrium the mass oscillates, and is determined by how far the mass was initially displaced from its equilibrium position. How long it takes for the object to complete one cycle of these periodic oscillations is called the period, T, is determined by the stiffness of the spring (described by the spring constant k) and the amount of mass m that is hanging from it. You can use the equations in the figure above to relate the period of motion to the frequency, f, and angular frequency, ?, of the oscillations. By solving Newton's Second Law we can find an equation that describes this periodic motion as a function of time. By taking the initial equilibrium position of the hanging mass-spring system to be y = 0 (previously yeq in Lab 6) the equation can be written as follows: y(t) = A sin(?t) To practice with a specific example, consider an oscillating spring mass system that is described by the following equation of motion: y(t) = 0.2 sin(9.4 t) 3. From this equation, what is the full distance between the highest and lowest points in the trajectory of the oscillating mass? (Hint: Think about how amplitude is defined.) 4. What is the frequency of oscillations, f, of the system? (Report your answer in units of Hz.)

Simple Harmonic Oscillations

As it turns out, in an ideal system with no friction or air resistance, the object will go on oscillating up and down forever. This kind of motion is called simple harmonic motion or simple harmonic oscillations.

The periodic trajectory of the oscillating mass over time is shown in the figure above. The amplitude of the oscillations, A, describes how far away from equilibrium the mass oscillates, and is determined by how far the mass was initially displaced from its equilibrium position. How long it takes for the object to complete one cycle of these periodic oscillations is called the period, T, is determined by the stiffness of the spring (described by the spring constant k) and the amount of mass m that is hanging from it. You can use the equations in the figure above to relate the period of motion to the frequency, f, and angular frequency, ?, of the oscillations.

By solving Newton's Second Law we can find an equation that describes this periodic motion as a function of time. By taking the initial equilibrium position of the hanging mass-spring system to be y = 0 (previously yeq in Lab 6) the equation can be written as follows:

y(t) = A sin(?t)

To practice with a specific example, consider an oscillating spring mass system that is described by the following equation of motion:

y(t) = 0.2 sin(9.4 t)

3. From this equation, what is the full distance between the highest and lowest points in the trajectory of the oscillating mass? (Hint: Think about how amplitude is defined.)

4. What is the frequency of oscillations, f, of the system? (Report your answer in units of Hz.)

Simple Harmonic Oscillations

As it turns out, in an ideal system with no friction or air resistance, the object will go on oscillating up and down forever. This kind of motion is called simple harmonic motion or simple harmonic oscillations.

The periodic trajectory of the oscillating mass over time is shown in the figure above. The amplitude of the oscillations, A, describes how far away from equilibrium the mass oscillates, and is determined by how far the mass was initially displaced from its equilibrium position. How long it takes for the object to complete one cycle of these periodic oscillations is called the period, T, is determined by the stiffness of the spring (described by the spring constant k) and the amount of mass m that is hanging from it. You can use the equations in the figure above to relate the period of motion to the frequency, f, and angular frequency, ?, of the oscillations.

By solving Newton's Second Law we can find an equation that describes this periodic motion as a function of time. By taking the initial equilibrium position of the hanging mass-spring system to be y = 0 (previously yeq in Lab 6) the equation can be written as follows:

y(t) = A sin(?t)

To practice with a specific example, consider an oscillating spring mass system that is described by the following equation of motion:

y(t) = 0.2 sin(9.4 t)

3. From this equation, what is the full distance between the highest and lowest points in the trajectory of the oscillating mass? (Hint: Think about how amplitude is defined.)

4. What is the frequency of oscillations, f, of the system? (Report your answer in units of Hz.)

Added by David E.

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