PHYS 1C Lecture Notes - Lecture 3: Simple Harmonic Motion, Equilibrium Point

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11 Apr 2018

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(3) lecture 12b simple harmonic motion and energy. Initial conditions at t=0 are: x(0) = 0 and v(0) = vi. The graph is shifted one quarter cycle to the right compared to the graph of x(0) = a. Let"s assume a spring-mass system is moving on a frictionless horizontal surface. This is an isolated system, therefore the total energy of the system is constant. Let"s assume a block moves to the right (+x) at an initial velocity, vi, on a frictionless surface toward an unstretched spring with spring constant, k: Initially, the total mechanical energy of the system is entirely due to the kinetic energy of the block: The total mechanical energy is now shared between the kinetic energy of the block and the spring week 1 page 1. Now, a little time later, the block has partially compressed the spring a distance x and has slowed to a velocity, v.

Related Questions

A particle of mass m is attachedbetween two identical springs on a horizontal frictionless table.Both springs have spring constant k and are initially un-stretched.If the particle is pulled a distance x along a directionperpendicular to the initial configuration of the springs, theforce exerted by the springs on the particle is given by F =-2kx(1-(L/sqrt(x^2+L^2)))

a) determine the amount of work done by this force in moving theparticle from x = A to x = 0.

b) show that the potential energy of the system is U(x) = kx^2 +2kL(L - sqrt(x^2 + L^2))

c) make a plot of U(x) versus x and identify all equilibriumpoints.

d) assume that m = 1.81 kg, L = 1.20 m and k = 40.0 N/m. If theparticle is pulled 0.500 m to the right and then released, what isits speed when it reaches the equilibrium point x = 0?

A 12-kg block on a horizontal frictionless surface is attached to a light spring (force constant = 0.80 kN/m). The block is initially at rest at its equilibrium position when a force (magnitude P= 80 N) acting parallel to the surface is applied to the block, as shown. What is the speed of the block when it is 13 cm from its equilibrium position?

Given data:

Mass of the block, \[m=12\text{ kg}\]

Spring constant, \[k=0.80\text{ kN/m = 800 N/m}\]

Force, \[F=80\text{ N}\]

Elongation in the spring, \[x=13\text{ cm = 0}\text{.13 m}\]

Applying the conservation of energy:

\[\begin{align}

& F.x=\frac{1}{2}k{{x}^{2}}+\frac{1}{2}m{{v}^{2}} \\

& \therefore \\

& {{v}^{2}}=\frac{2\left( F.x-\frac{1}{2}k{{x}^{2}} \right)}{m} \\

& v=\sqrt{\frac{2\left( F.x-\frac{1}{2}k{{x}^{2}} \right)}{m}}

\end{align}\]

Plugging in the given values:

\[\begin{align}

& v=\sqrt{\frac{2\left( 80\times 0.13-\frac{1}{2}\times 800\times {{\left( 0.13 \right)}^{2}} \right)}{12}} \\

& =0.7788\text{ m/s}

\end{align}\]

Answer:

Speed of the block: 0.778 m/s

greylocust991

a).An object of mass 541 g is attached to a spring with springconstant k = 653 N/m. If the object is displaced by 130 mm from theequilibrium position and is released, with what speed doe theobject pass through the equilibrium position during itsmotion?

Express the result in the unit [m/s] and to three significantfigures. If you must use scientific notation, please enter asfollows: e.g. 0.000123 = 1.23E-4. Only answer in numerical values,no units. For negative numbers, leave no space between the negativesign and the number: e.g. right: -1.00, wrong: - 1.00.

b).An object has a mass of m = 647 g. It is attached to a springwhich has a spring constant of value k = 65 N/m. At time t = 0 theobject is pulled to a distance of 28.1 cm from its equilibriumposition (which you may choose conveniently at x = 0). The surfaceon which the object moves is frictionless.

What is the total energy of the system?

Express the result in the unit [J] and to three significantfigures. If you must use scientific notation, please enter asfollows: e.g. 0.000123 = 1.23E-4. Only answer in numerical values,no units. For negative numbers, leave no space between the negativesign and the number: e.g. right: -1.00, wrong: - 1.00.

c).The vibration of the tympanic membrane in the human ear(eardrum) can be described as a harmonic motion. Assuming a mass of20.6 mg and a period of vibration of 1.72 ms, what is the naturalvibration frequency of the membrane?

Express the result in the unit [Hz] and to three significantfigures. If you must use scientific notation, please enter asfollows: e.g. 0.000123 = 1.23E-4. Only answer in numerical values,no units. For negative numbers, leave no space between the negativesign and the number: e.g. right: -1.00, wrong: - 1.00.

d).An object of mass m is attached to a spring with spring constantk = 170 N/m. At time t = 0 the object is pulled to a distance of0.18 m from its equilibrium position. The surface on which theobject moves is frictionless and the spring is within its elasticregime. What is the mass of the object if it oscillates with afrequency f = 2.90 Hz when it is released?

Express the result in the unit [kg] and to three significantfigures. If you must use scientific notation, please enter asfollows: e.g. 0.000123 = 1.23E-4. Only answer in numerical values,no units. For negative numbers, leave no space between the negativesign and the number: e.g. right: -1.00, wrong: - 1.00.

PHYS 1C Lecture Notes - Lecture 3: Simple Harmonic Motion, Equilibrium Point

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