A mass m=4.5 kg is attached to a vertical spring with k=200...

A mass $m=4.5 \mathrm{kg}$ is attached to a vertical spring with $k=200 \mathrm{N} / \mathrm{m}$ and is set into motion. a. What is the frequency of the oscillation? b. If the amplitude of the oscillation is $3.5 \mathrm{cm}$, what is the maximum value of the velocity? c. How long does it take the mass to move from $y=1.5 \mathrm{cm}$ to $y=2.5 \mathrm{cm} ?$ d. If the mass is oscillating with a maximum speed of $45 \mathrm{m} / \mathrm{s}$, what is the amplitude? e. If the spring constant is increased by a factor of two and the maximum kinetic energy of the mass is the same, by what factor does the amplitude change?

A mass $m=4.5 \mathrm{kg}$ is attached to a vertical spring with $k=200 \mathrm{N} / \mathrm{m}$ and is set into motion.
a. What is the frequency of the oscillation?
b. If the amplitude of the oscillation is $3.5 \mathrm{cm}$, what is the maximum value of the velocity?
c. How long does it take the mass to move from $y=1.5 \mathrm{cm}$ to $y=2.5 \mathrm{cm} ?$
d. If the mass is oscillating with a maximum speed of $45 \mathrm{m} / \mathrm{s}$, what is the amplitude?
e. If the spring constant is increased by a factor of two and the maximum kinetic energy of the mass is the same, by what factor does the amplitude change?

Key Concepts

Simple Harmonic Motion

This is a type of periodic motion where the restoring force is directly proportional to the negative of the displacement, typically observed in mass?spring systems. The motion is described by sinusoidal functions and features characteristics such as period, frequency, amplitude, and phase, which together fully describe the oscillation.

Angular Frequency

Angular frequency, often denoted as ?, is a fundamental quantity in oscillatory systems defined as the rate at which the system cycles through its motion in radians per unit time. In mass?spring systems, it is given by ? = ?(k/m), linking the system’s mass and the stiffness of the spring, and it directly determines the oscillation period and the frequency of the motion.

Amplitude

Amplitude represents the maximum displacement from the equilibrium position in oscillatory motion. It is a key parameter that determines the overall energy of the oscillation and, when combined with angular frequency, helps define other characteristics such as maximum velocity and maximum kinetic energy.

Maximum Speed in Simple Harmonic Motion

In simple harmonic motion, the maximum speed of the oscillating object occurs as it passes through the equilibrium position. This maximum speed is directly proportional to both the amplitude and the angular frequency, typically given by the relation v_max = A·?, which is crucial in connecting motion parameters with energy considerations.

Kinetic and Potential Energy in Oscillatory Systems

The interplay of kinetic and potential energy characterizes simple harmonic motion. At maximum displacement, the energy is entirely potential (in the spring), while at the equilibrium position, it is entirely kinetic. The conservation of energy in these systems can be used to relate amplitude, mass, and angular frequency, especially when comparing different configurations or conditions.

Phase and Time Calculations in SHM

The sinusoidal nature of simple harmonic motion means that any displacement can be related to a phase angle within the cycle. By inverting these trigonometric relationships, one can determine the time intervals required for the object to move between specific positions, an essential method for analyzing the timing aspects of oscillatory motion.

Effects of Changing System Parameters

Modifications to system parameters, such as a change in the spring constant, affect the angular frequency and subsequently the entire dynamic behavior of the system. For example, if the spring constant increases while maintaining other conditions such as kinetic energy, the amplitude must adjust to balance the energy relations, highlighting the interdependent nature of the physical parameters in oscillatory systems.

Transcript

Please give Ace some feedback