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2 years ago

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A spherical shell has an outside radius of 2.60 cm and an inside radius of a. The shell wall has uniform thickness and is made o

f a material with density 4.70 g/cm3. The space inside the shell is filled with a liquid having a density of 1.23 g/cm3. (a) Find the mass m of the sphere, including its contents, as a function of a. (b) In the answer to part (a), if a is regarded as a variable, for what value of a does m have its maximum possible value

1 answer:

lbvjy [14]2 years ago

4 0

Answer:

The answer is below

Explanation:

a) The volume of a sphere is:

Volume = (4/3)πr³; where r is the radius of the shell.

Given the outside radius of 2.60 cm and inner radius of a cm, the volume of the spherical shell is:

Volume of spherical shell = $\frac{4}{3} \pi (2.6^3-a^3)$ cm³

But Density = mass / volume; Mass = density * volume.

Therefore, mass of spherical shell = density * volume

mass of spherical shell = $4.70\ g/cm^3$ * $\frac{4}{3} \pi (2.6^3-a^3)$ cm³

Mass of liquid = volume of inner shell * density of liquid

Mass of liquid = $\frac{4}{3} \pi a^3\ cm^3*1.23\ g/cm^3$

Total mass of sphere including contents = mass of spherical shell + mass of liquid

Total mass of sphere including contents (M) = $4.70\ g/cm^3$ * $\frac{4}{3} \pi (2.6^3-a^3)\ cm^3$ + $\frac{4}{3} \pi a^3\ cm^3*1.23\ g/cm^3$ =

Total mass of sphere including contents (M) = (346 - 14.5a³) grams

b) The mass is maximum when the value of a = 0

M = 346 - 14.5a³

M' = 43.5a² = 0

43.5a² = 0

a = 0

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Answer:

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Therefore, at maximum displacement, velocity is 0.

Practically, this can be observed in a simple pendulum. As it approaches the maximum displacement, its velocity reduces. It becomes zero at this point and then reverses as the pendulum changes course. Then the velocity begins to increase. It becomes maximum at the equilibrium point but once past that, the velocity begins to reduce as it approaches the other amplitude.

For acceleration,

$a = -\omega^2x$

It follows that at maximum displacement, the acceleration is a maximum. The negative sign indicates that it is in an opposite direction to the displacement. Both kinetic energy ( $\frac{1}{2}mv^2$ ) and linear momentum ( $mv$ ) are proportional to velocity; they are therefore both zero at the maximum displacement.

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A mass weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the mass is initially relea

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A mass weighing 32 pounds stretches a spring 2 feet.

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(b) How many complete cycles will the mass have completed at the end of 4 seconds?

Answer:

$A = 1.803 ft$

Period = $\frac{\pi}{2}$ seconds

8 cycles

Explanation:

A mass weighing 32 pounds stretches a spring 2 feet;

it implies that the mass (m) = $\frac{w}{g}$

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Also from Hooke's Law

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Using the function:

$\frac{d^2x}{dt} = - 16x\\\frac{d^2x}{dt} + 16x =0$

$x(0) = -1$ (because of the initial position being above the equilibrium position)

$x(0) = -6$ ( as a result of upward velocity)

NOW, we have:

$x(t)=c_1cos4t+c_2sin4t\\x^{'}(t) = 4(-c_1sin4t+c_2cos4t)$

However;

$x(0) = -1$ means

$-1 =c_1\\c_1 = -1$

$x(0) =-6$ also implies that:

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Hence, $x(t) =-cos4t-\frac{3}{2} sin 4t$

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$A = 1.803 ft$

Period can be calculated as follows:

= $\frac{2 \pi}{4}$

= $\frac{\pi}{2}$ seconds

How many complete cycles will the mass have completed at the end of 4 seconds?

At the end of 4 seconds, we have:

$x* \frac{\pi}{2} = 4 \pi$

$x \pi = 8 \pi$

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Answer:

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