PHYS 1501Q Chapter Notes - Chapter 18: Phase Angle, Simple Harmonic Motion, Equilibrium Point

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Apply newton"s 2nd law to the system. Procedure to determine amplitude and phase of the motion. Spring and mass are free to oscillate. Mass and spring at rest at x = 0. Emechanical = is conservative (and is the only force doing work on the spring) . K constant is at maximum, is at maximum, U spring is at minimum is at minimum (newton"s 2nd law) (and therefore a. ) are not constant so kinematics equations cannot be used. A = dt d x2 = k. Sines and cosines d x2 = 2 dt2 sin( t) x. Distinguishment of cosine or sine graph depends on where t=0 is placed on the graph. 2 x = r cos( t) y = r sin( t) A is maximum displacement or radius is if the first occurrence does not occur at t=0 or initial angular displacement. The mass is pulled a distance +d and then released from rest at t=0.

Related Questions

A block of mass m is attached to a spring whose spring constant is k. The other end of the spring is fixed so that when the spring is unstretched, the mass is located at x=0. . Assume that the +xdirection is to the right. The mass is now pulled to the right a distance A beyond the equilibrium position and released, at time t=0, with zero initial velocity. Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion.

the system will undergo simple harmonic motion. For such a system, the equation of motion is $a(t)=-\frac{k}{m}x(t)$ ,

and its solution, which provides the equation for , is $x(t)=A\cos\left(\sqrt{\frac{k}{m}}t\right)$ .

ParA: At what time t_1

does the block come back to its original equilibrium position ( $x=0$ ) for the first time?

Express your answer in terms of some or all of the variables:

, and

Part B

Find the velocity

of the block as a function of time.

Express your answer in terms of some or all of the variables:

, and

Part C

What is the maximum speed v_max

of the block during this motion?

Express your answer in terms of some of all of the variables:

, and

olivewater-buffalo778

Good Vibes: Introduction to Oscillations

The conditions that lead to simple harmonic motion are as follows:

There must be a position of stable equilibrium.

There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by ⃗, where is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

Consider a block of mass m attached to a spring with force constant k, as shown in the figure(Figure 1) . The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from x=A, it will

(a) remain at rest

(b) move to the left until it reaches equilibrium and stop there.

(d) move to the left until it reaches x=−A and then begin to move to the right.

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.

The frequency, denoted f, is the number of cycles that are completed per unit of time: f=1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).

Part B

If the period is doubled, the frequency is

(a) unchanged

(b) doubled

Part C

An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?

Express your answer in hertz.

Part D

If the frequency is 40 Hz, what is the period T?

Express your answer in seconds.

The following questions refer to the figure (Figure 2) that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x-coordinate of the oscillating object, and the horizontal axis represents time.

Part E

Which points on the x-axis are located at a distance A from the equilibrium position?

(a) R only

(b) Q only

Part F

Suppose that the period is T. Which of the following points on the t-axis are separated by the time interval T?

(a) K and L

(b) K and M

(d) L and N

(e) M and P

Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.

Part G

What is the period T?

Express your answer in seconds.

Part H

How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?

Express your answer in seconds.

Part I

What distance d does the object cover during one period of oscillation?

Express your answer in meters.

Part J

What distance d does the object cover between the moments labeled K and N on the graph?

Express your answer in meters.

emeraldyak680

particleof mass m = 1.18 kg is attached between two identical springs on ahorizontal frictionless tabletop. The springs have spring constantk, and each is initially unstressed. (a) If the mass of pulled adistance x along a direction perpendicular to the initialconfiguration of the springs, show that the potential energy of thesystem is:
U(x)=K(X^2) + 2KL (L- square-root(X^2 + L^2) {look above thequestion for better viewing equation }
(b) Make a plot of U(x) versus x and identify all equilibriumpoints.
Assume that L = 1.20 m and k = 40.0 N/m. (c) If the mass is
pulled 0.500 to the right and then released, what is its speed whenit
reaches the equilibrium point x = 0? (d) Using the potentialenergy
of the system find the force as a function of x.

NOTE: the picture is top view pictute

PHYS 1501Q Chapter Notes - Chapter 18: Phase Angle, Simple Harmonic Motion, Equilibrium Point

PHYS 1501Q Full Course Notes

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