Physics, 26.11.2020 01:00 noreenhussain

A mass m at the end of a spring of spring constant k is undergoing simple harmonic oscillations with amplitude A. Part (a) At what positive value of displacement x in terms of A is the potential energy 1/9 of the total mechanical energy?
Part (b) What fraction of the total mechanical energy is kinetic if the displacement is 1/2 the amplitude?
Part (c) By what factor does the maximum kinetic energy change if the amplitude is increased by a factor of 3?

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Answer from: 22iungj

a) The potential energy of the system is 1/9 of the total mechanical energy, when $x= \frac{1}{3}\cdot A$ .

b) The fraction of the total mechanical energy that is kinetic if the displacement is 1/2 the amplitude is 1/2.

c) The maximum kinetic energy is increased by a factor of 9.

Explanation:

a) From Mechanical Physics, we remember that the mechanical energy of mass-spring system (), measured in joules, is the sum of the translational kinetic energy (), measured in joules, and elastic potential energy (), measured in joules. That is:

E = K + U (1)

By definitions of translational kinetic energy and elastic potential energy, we have the following expressions:

$K = \frac{1}{2}\cdot m \cdot v^{2}$ (2)

$U = \frac{1}{2}\cdot k\cdot x^{2}$ (3)

Where:

- Mass, measured in kilograms.

- Velocity of the mass, measured in meters per second.

- Spring constant, measured in newtons per meter.

- Elongation of the spring, measured in meters.

If we know that $U = \frac{1}{9}\cdot E$ , k = k and $E = \frac{1}{2}\cdot k \cdot A^{2}$ , then:

$\frac{1}{18}\cdot k\cdot A^{2} = \frac{1}{2}\cdot k\cdot x^{2}$

$\frac{1}{9}\cdot A^{2} = x^{2}$

$x= \frac{1}{3}\cdot A$

The potential energy of the system is 1/9 of the total mechanical energy, when $x= \frac{1}{3}\cdot A$ .

b) If we know that k = k , $x = \frac{1}{2}\cdot A$ and $E = \frac{1}{2}\cdot k \cdot A^{2}$ , then the equation of energy conservation associated with the system is:

$\frac{1}{2}\cdot k\cdot A^{2} = \frac{1}{4}\cdot k\cdot A^{2}+K$

$K = \frac{1}{4}\cdot k\cdot A^{2}$

The fraction of the total mechanical energy that is kinetic if the displacement is 1/2 the amplitude is 1/2.

c) From the Energy Conservation equation associated with the system, we know that increasing the amplitude by a factor of 3 represents an increase in the elastic potential energy by a factor of 9. Then, the maximum kinetic energy is increased by a factor of 9.

Answer from: Quest

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Answer from: Quest

The boat's velocity relative to the riverbank can be resolved into two components. the component going north is 2.7 m/s and the component going west is 1.2 m/s.we add these two vectors and then find the magnitude of the resultant to get the speed.the magnitude of the resultant vector from adding can be obtained by the pythagorean theorem. $\begin{aligned} | \vec{v}| & = \sqrt{(\text{2.7 m/s})^2 + (\text{1.2 m/s})^2 } \\ & = 2.954 \text{ m/s} \end{aligned}$ to two significant figures, the boat's speed relative to the riverbank is 3.0 m/s.

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