Physics, 15.01.2020 01:31 coopera1744

Asolid, uniform disk of mass m and radius a may be rotated about an axis parallel to the disk axis, at variable distances from the center of the disk.
what is tmin, the minimum period of the pendulum?

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The minimum period of the pendulum Tmin is $\mathbf{T_{min} = 2 \pi \sqrt{\dfrac{a^2}{ \Big ( \dfrac{ga}{\sqrt{2}}\Big)}} \ \ or \ \ 2 \pi\sqrt{ \dfrac{\sqrt{2a}}{g}}}$

The period of a pendulum refers to the time required to complete one single revolution or cycle.

Let consider the time period of the disk to that of bob of a pendulum, then, the period of the pendulum can be expressed as:

$\mathbf{T(d) = 2 \pi \sqrt{\dfrac{a^2}{2gd}+\dfrac{d}{g} }}$

At minimum, the period of the pendulum needs to obey the condition that:

$\mathbf{\dfrac{dT}{d(d)}=0}$

∴

The above equation becomes:

$\mathbf{\dfrac{d \Big( 2 \pi \sqrt{\dfrac{a^2}{2gd}+\dfrac{d}{g} }\Big )}{d(d)}=0}$

Taking the differentiation of the above equation with respect to d, we have:

$\mathbf{d = \dfrac{a}{\sqrt{2}}}$

∴

$\mathbf{T_{min} = 2 \pi \sqrt{\dfrac{a^2}{2g \Big ( \dfrac{a}{\sqrt{2}} \Big)} + \dfrac{a}{g \sqrt{2}}}}$

$\mathbf{T_{min} = 2 \pi \sqrt{\dfrac{a^2}{ \Big ( \dfrac{ga}{\sqrt{2}}\Big)}} \ \ or \ \ 2 \pi\sqrt{ \dfrac{\sqrt{2a}}{g}}}$

Learn more about the period of a pendulum here:

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Answer from: Merida06

the function varies linearly with the radius of the disk, so the smallest period is zero for a radius of zero centimeters

Explanation:

This system performs a simple harmonic movement where the angular velocity is given by

w = √ k / I

Where k is the constant recovered from the axis of rotation and I is the moment of inertia of the disk

The expression for the moment of inertia is

I = 1/2 m r²

Angular velocity, frequency and period are related

w = 2π f = 2π / T

Substituting

2π / T = √ k / I

T = 2π √ I / k

T = 2π √ (½ m r² / k)

T = (2π √m / 2k) r

We can see that the function varies linearly with the radius of the disk, so the smallest period is zero for a radius of zero centimeters

Answer from: Quest

an ion is a charged atom or molecule. it is charged because the number of electrons do not equal the number of protons in the atom or molecule. an atom can acquire a positive charge or a negative charge depending on whether the number of electrons in an atom is greater or less then the number of protons in the atom.

Answer from: Quest

i dont think you are wrong for wanting to get even with someone who did you wrong.

(i'm an eye for an eye kind of person) but if this person hurt you in a sexual way, i dont suggest doing anything to him. he should be taken to the proper authorities, you could go to jail depending on what you do. if it was something small like your sibling took your spot on the couch, than yea, i would get even with him by taking one of his favorite snacks lol. but i dont suggest you doing anything illegal.

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