a speeding car traveling north at 80mph passes the tramway intersection when the police begin pursuit. The police start 6 miles south of tramway traveling at 100mph. How many miles north of tramway will they be when they catch up with the speeding car?

Final answer:

The density of a rectangular solid can be determined in a lab by finding the mass and volume of the object and dividing them. The mass is measured using an analytical balance, and the volume is calculated from the geometric parameters.

Explanation:

The density of a rectangular solid can be determined in a lab by separately finding the mass and volume of the object and then dividing the mass by the volume. The mass can be measured using an analytical balance, while the volume can be calculated from the geometric parameters. For example, the volume of a rectangle is equal to length x width x height, and the volume of a cube is equal to the edge length cubed.

Let's say we have a rectangular solid with a length of 10 cm, a width of 5 cm, and a height of 2 cm. To determine the density of this solid, we would first measure its mass using an analytical balance. Let's assume the mass is 100 grams. Next, we would calculate the volume of the rectangular solid by multiplying its length, width, and height together: 10 cm x 5 cm x 2 cm = 100 cm³. Finally, we would divide the mass by the volume to find the density: 100 g / 100 cm³ = 1 g/cm³.

To stay submerged in salt water, a 53.0 kg grouper with a body density of 1013 kg/m3 must exert a force of approximately 7.27 N. This is calculated using the principle of buoyancy to determine the buoyant force in relation to the grouper's weight.

The force a 53.0 kg grouper must exert to stay submerged in salt water with a body density of 1013 kg/m3 can be found by applying the principle of buoyancy (Archimedes' principle), which states that the buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object.

First, calculate the volume of the grouper. Since density ( ) equals mass (m) divided by volume (V), we have V = m / . For a 53.0 kg grouper with a density of 1013 kg/m³, the volume V would be 53.0 kg / 1013 kg/m³ = 0.05234 m³.

Next, calculate the weight of the volume of salt water displaced. The density of salt water is approximately 1027 kg/m3. The weight of the displaced water (Ww) is the product of its volume (V), its density ( water), and the acceleration due to gravity (g). So, Ww = V × water × g = 0.05234 m³ × 1027 kg/m³ × 9.81 m/s² = 527.3 N.

Finally, the force the grouper must exert to stay submerged (F) is the difference between the buoyant force and the grouper's weight. The weight of the grouper (Wg) is calculated as mass times gravitational acceleration, Wg = 53.0 kg ×9.81 m/s2 = 520.03 N. Thus, F = Ww - Wg = 527.3 N - 520.03 N = 7.27 N.

Therefore, a 53.0 kg grouper must exert a force of approximately 7.27 N to stay submerged in salt water.

The grouper must exert a force of approximately [tex]\( 516.88 \, \text{N} \)[/tex] to stay submerged in salt water.

To determine the force that the grouper must exert to stay submerged in salt water, we can use Archimedes' principle, which states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

The buoyant force [tex](\( F_b \))[/tex] can be calculated using the formula:

[tex]\[ F_b = V \times \rho_{\text{fluid}} \times g \][/tex]

Where:

[tex]\( V \)[/tex] is the volume of the object submerged in the fluid

[tex]\( \rho_{\text{fluid}} \)[/tex] is the density of the fluid

[tex]\( g \)[/tex] is the acceleration due to gravity

The weight of the grouper [tex](\( F_g \))[/tex] can be calculated using the formula:

[tex]\[ F_g = m \times g \][/tex]

For the grouper to stay submerged, the buoyant force must be equal to the weight of the grouper. Therefore:

[tex]\[ F_b = F_g \][/tex]

[tex]\[ V \times \rho_{\text{fluid}} \times g = m \times g \][/tex]

[tex]\[ V \times \rho_{\text{fluid}} = m \][/tex]

[tex]\[ V = \frac{m}{\rho_{\text{fluid}}} \][/tex]

Now, we can calculate the volume of the grouper submerged in the fluid:

[tex]\[ V = \frac{53.0 \, \text{kg}}{1013 \, \text{kg/m}^3} \][/tex]

[tex]\[ V = \frac{53.0 \, \text{kg}}{1013 \, \text{kg/m}^3} \][/tex]

[tex]\[ V = 0.05236 \, \text{m}^3 \][/tex]

Now, we can use this volume to calculate the buoyant force:

[tex]\[ F_b = V \times \rho_{\text{fluid}} \times g \][/tex]

[tex]\[ F_b = 0.05236 \, \text{m}^3 \times 1013 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 \][/tex]

[tex]\[ F_b = 516.88 \, \text{N} \][/tex]

Therefore, the grouper must exert a force of approximately [tex]\( 516.88 \, \text{N} \)[/tex] to stay submerged in salt water.

Final answer:

Galileo most likely used a water clock or a pendulum clock to measure the time objects took to hit the ground. The mass of the objects does not affect the time it takes for them to fall. On the Moon, the time it takes for objects to hit the ground would be different, but the ratio of their times would remain the same.

Explanation:

When Galileo conducted his experiment of dropping two objects of different masses from the Tower of Pisa, he most likely used a water clock or a pendulum clock to measure the time it took for each object to reach the ground. Although stopwatches weren't available at that time, water clocks and pendulum clocks were commonly used as timekeeping devices.

If the objects were the same size but had different masses, Galileo should have observed that both objects hit the ground at the same time. This is because, in the absence of air friction, all objects experience the same acceleration due to gravity. Hence, the difference in mass does not affect the time it takes for an object to fall.

If the experiment were done on the Moon, where the acceleration due to gravity is approximately one-sixth of that on Earth, the time it takes for the objects to hit the ground would be different. However, the ratio of their times would remain the same, meaning that the second rock would still need to be dropped 0.50 s after the first rock to hit the ground simultaneously.

Since the index of refraction of the film is larger than that of air (n = 1) there is an additional phase shift for the reflection in the soap film.

The formula for constructive interference is

2L = (m+ 0.5)λ/n

Where,

L = thickness of the film

m = order number

λ = wavelength

n = index of refraction = 1.50

Rewriting in terms of λ:

 λ = 3L/(m+ 0.5)

The information that λ = 800 nm and λ = 480 nm are consecutive maximum means that if λ = 800 nm refers to m, then λ = 480 nm refers to m + 1. Using the m dependence on λ, this implies that:

800 / (m + 1 + 0.5) = 480 / (m + 0.5)

800 (m + 0.5) = 480 (m + 1.5)

800 m + 400 = 480 m + 720

320 m = 320

m = 1

In other words for m = 1 and λ = 800 nm:

L = (m + 0.5)λ/3 = (1.5)*800/3 = 400 nm

The soap film is under constructive interference from the light. Given that the index of refraction of the film is n=1.5, the thinnest possible thickness of the film using the longest wavelength (800 nm) results in a film thickness of around 266.5 nm.

The phenomenon under study here is known as thin film interference. When light shines on a thin film like a soap bubble, some light is reflected from the top surface of the film, and some light is refracted and travels through the film and reflects off the bottom surface. These two rays of light can interfere constructively or destructively depending on the thickness of the film and the light's wavelength.

Given that only wavelengths of 480 nm and 800 nm are intensified, this indicates constructive interference - that is, the path difference between the two rays is a multiple of the wavelength. Because the soap film has an index of refraction of n = 1.5, the wavelength of light in the film will be the vacuum wavelength divided by n.

For constructive interference in a film, we have the condition that twice the film thickness equals m wavelengths in the film for some integer m. In other words, 2t = mλ’. To apply this condition, we use the longest wavelength (800 nm) to get the thinnest possible film thickness, since a larger m would imply a thicker film. From λ’ = λ/n = 800/1.5 ~ 533 nm, we have 2t = λ’, which means t = λ’/2 ~ 533/2 = 266.5 nm.

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Yes, a test could be performed to support the claim.

Hypothesis: The claim that a manufacturer’s cleanser works twice as fast as any other cleanser.

So, based from this hypothesis, we can perform the following tests:

We assign Cleanser A to the manufacturer that claims that their cleanser works twice as fast as any other cleanser and Cleanser B to the cleanser to be compared with.

1.       Get two tiles and put the same amount of stain on them.

2.       Apply Cleanser A on the first tile and Cleanser B on the second tile.

3.       Apply the same amount of force in removing the stains on both tiles

4.       Record the amount of time it takes to remove the stains on each tile.

The acceleration of a 300,000-kg jumbo jet with a thrust force of 120,000 N is calculated using Newton's second law of motion to be 0.4 m/s².

The student has asked a Physics question related to calculating the acceleration of an object given its mass and the force applied to it. The subject of this question falls under Newton's second law of motion, which states that the acceleration (a) of an object is directly proportional to the net force (F) acting on it and inversely proportional to its mass (m), which can be represented by the equation a = F / m.

In the case of the jumbo jet with a mass of 300,000 kg experiencing a thrust force of 120,000 N, we can find the acceleration of the jet by using Newton's second law:

a = F / m = 120,000 N / 300,000 kg = 0.4 m/s²

The acceleration of the jumbo jet is 0.4 m/s² just before takeoff.

The electric field midway between the two point particles, one with charge +8 × 10⁻⁹ c and the other with charge -2 × 10⁻⁹ c, separated by 4 m, is 1124 N/C, directed away from the positive charge.

The electric field at a point is the force that a unit of positive charge would experience if placed at that point. It is given by the Coulomb's Law formula, E = k*q/r², where k is Coulomb's constant (8.99 x 10⁹ N.m²/C²), q is the charge and r is the distance from the charge.

For the given question, the two point charges are +8 × 10⁻⁹ C and -2 × 10⁻⁹ C. The point where we need to find the electric field is midway, so the distance from each charge is 2m. The directions of the electric fields due to the positive and negative charges are opposite at this point.

Calculating the electric field caused by each charge: For positive charge (E₁): E₁ = kxq₁/r₁² = (8.99 x 10⁹ N.m²/C²)x(8 × 10⁻⁹ C)/(2 m)² = 899 N/C, and for the negative charge (E₂): E₂ = kxq₂/r₂² = (8.99 x 10⁹ N.m²/C²)x(-2 × 10⁻⁹ C)/(2 m)² = -225 N/C.

The resultant electric field E at the midpoint is the vector sum of E₁ and E₂. As they are directed in opposite directions, we subtract E₂ from E₁, giving E = E₁ - E₂ = 899 N/C - (-225 N/C) = 1124 N/C, directed away from the positive charge.

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1 slug = 32 lb 

f = kx 
32 = k(2) 
k = 16 

c = 8 ( 8 times the instantaneous velocity) 

mx'' + cx' + kx = 8sin4t 
x'' + 8x' + 16x = 8sin4t 

Find for the complimentary solution xh:
r² + 8r + 16 = 0 
r² + 4r + 4r + 16 = 0 
(r + 4)(r + 4) = 0 
r = -4, -4 (repeated roots) 
xh = c₁e^(-4t) + c₂te^(-4t) 

Find for the particular solution xp:
xp = Acos(4t) + Bsin(4t) 
xp' = -4Asin(4t) + 4Bcos(4t) 
xp'' = -16Acos(4t) - 16Bsin(4t) 
x'' + 8x' + 16x = 8sin(4t) 
-16Acos(4t) - 16Bsin(4t) + 8[ -4Asin(4t) + 4Bcos(4t) ] + 16 [ Acos(4t) + Bsin(4t) ] = 8sin(4t) 
-16Acos(4t) - 16Bsin(4t) - 32Asin(4t) + 32Bcos(4t) + 16Acos(4t) + 16Bsin(4t) ] = 8sin(4t) 
-32Asin(4t) + 32Bcos(4t) = 8sin(4t) 
-4Asin(4t) + 4Bcos(4t) = sin(4t) 

We group like terms and then solve for A and B:
4Bcos(4t) = 0 
B = 0 

-4Asin(4t) + 4Bcos(4t) = sin(4t) 
-4Asin(4t) = sin(4t) 
A = -¼ 

xp = Acos(4t) + Bsin(4t) 
xp = -¼cos(4t) + (0) sin(4t) 
xp = -¼cos(4t) 

The general solution is therefore: 
x(t) = xh + xp 
x(t) = c₁e^(-4t) + c₂te^(-4t) - ¼ cos(4t) 

at t = 0 it starts from rest that is initial velocity = 0 
x'(0) = 0 

at t = 0 it starts from equilibrium 
x(0) = 0 

x(t) = c₁e^(-4t) + c₂te^(-4t) - ¼cos(4t) 
0 = c₁ + c₂(0) - ¼cos(0) 
c₁ = ¼ 

x(t) = c₁e^(-4t) + c₂te^(-4t) - ¼cos(4t) 
x(t) =¼e^(-4t) + c₂te^(-4t) - ¼cos(4t) 
x '(t) = -e^(-4t) + [ -4c₂te^(-4t) + c₂e^(-4t) ] + sin(4t) 
x '(t) = -e^(-4t) - 4c₂te^(-4t) + c₂e^(-4t) + sin(4t) 

x'(0) = 0 
0 = -e^(0) - 4c₂(0) e^(0) + c₂e^(0) + sin(0) 
0 = -1 + c₂ + 
= -4c₁ - 4c₂(0) + c₂ 
0= -4(1/4) + c₂ 
c₂ = 1 

x(t) =¼e^(-4t) + c₂te^(-4t) - ¼cos(4t) 
x(t) =¼e^(-4t) + te^(-4t) - ¼cos(4t) 

Answer:

temperature

Explanation:

The temperature of an object will automatically reflect the increase or decrease in the average kinetic energy of the molecules of the object, kinetic energy is related with the movement, but when the molecules of the object are moving and reflecting kinetic energy it is not necessary the case that that would be provoqued by the movement of the object so temperature would be the best way to measure the change in the molecules kinetic energy.

the color would be yellow..hope this helps :))

Because the two paths are perpendicular, therefore the target proton's new path must be at 30 degrees from the original direction. 

Using the law of conservation of momentum about the original direction: 
m (400 m/s) = m (v1) cos(60) + m (v2) cos(30) 
Cancelling m since the two protons have similar mass.
(v1)cos(60) + (v2)cos(30) = 500 m/s                         ---> 1

Now by using the law conservation of momentum perpendicular to the original direction: 
m (0 m/s) = m (v1) sin(60) – m (v2) sin(30) 
Which simplifies to:
(v1)sin(60) - (v2)sin(30) = 0 m/s                                
v2 = v1 * sin(60) / sin(30) = v1 * sqrt(3)                  ---> 2

Plugging equation 2 to equation 1: 
(v1) (1/2) + (v1 * sqrt(3)) sqrt(3)/2 = 500 m/s 
(1/2) (v1) + (3/2) (v1) = 500 m/s 
2 (v1) = 500 m/s 
v1 = 250 m/s 

Thus, from equation 2:

v2 = v1*sqrt(3) = (250 m/s) sqrt(3) = 433.01 m/s 

So,
A. The target proton's speed is about 433 m/s 
B. The projectile proton's speed is 250 m/s

The speed of the target proton and the projectile proton after the elastic collision are both 500 m/s.

For an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

Final answer:

The thinnest film that will produce constructive interference in the reflection of light with a wavelength of 510 nm for a film with a refractive index of 1.60 is 159.375 nm.

Explanation:

Thin Film Interference and Constructive Interference

To find the thinnest film that will produce constructive interference in the reflection of light with a wavelength of 510 nm for a film with n = 1.60, one can use the formula for constructive interference in thin films. The formula for the thinnest film thickness (t) for constructive interference, when light of wavelength λ in the film is incident normally, is given by:

t = (m λ) / (2n), where m is the order of the interference (m = 0, 1, 2, ...), λ is the wavelength of the light in vacuum, and n is the refractive index of the film.

For the first order of constructive interference (m=0), t should be minimum, so we use m = 0:

[tex]t = \frac{(0 \times 510 \ nm)}{(2 \times 1.60)} = 0 \ nm[/tex].

Since 0 nm doesn't represent a physical film, the next order (m=1) should be considered, so:

[tex]t = \frac{(1 \times 510 \ nm)}{(2 \times 1.60)} = 159.375 nm[/tex].

The minimum thickness for the first constructive interference is thus 159.375 nm.

Answer:

It increases the number of collision

The magnetic field at a point 1.2 m from the axis has a magnitude of 7.0 × 10^–6 T

Maxwell's equation is a set of coupled partial differential equations that together with the Lorentz force law form the classical electromagnetism, classical optics, and electric circuits.

Integral form in the absence of magnetic or polarizable media are Gauss' law for electricity, Gauss' law for magnetism, Faraday's law of induction, Ampere's law

A 0.70 m radius cylindrical region contains a uniform electric field that is parallel to the axis and is increasing at the rate [tex]5.0* 10^{12} v/m.s[/tex]  The magnetic field at a point 1.2 m from the axis has a magnitude of?

The Maxwell’s law of induction is as follows. Consider the charging of our circular  plate capacitor , B field also  induced at  point 2.  When capacitor stops charging  B field disappears.

By using the Maxwell’s law of induction for a circle of radius r.

[tex]2\pi rB = \epsilon_{0}\mu_{0}\pi r^2 \frac{dE}{dt} , B = \frac{1}{2}  \epsilon_{0}\mu_{0}r\frac{dE}{dt} = 7*10^{-6}T[/tex]

Grade: 9

Subject:  physics

Chapter:  electric field

Keywords: magnetic field, a uniform electric field, parallel,  the axis,  point

Final answer:

The question is related to Faraday's law, but cannot be answered without additional information regarding the changing electric field.

Explanation:

The question asks for the magnitude of the magnetic field at a point 1.2 m from the axis of a cylindrical region where there is a uniform electric field increasing at a given rate. This is related to Faraday's law of electromagnetic induction, which relates the time rate of change of the magnetic field to the induced electric field in the surrounding region. However, the question provided is incomplete and does not provide sufficient information to solve for the magnetic field at the given distance without information such as the direction or the specific distribution of the increasing electric field.

The speed obtained by the pilot is not accurate since it is measuring the rate of travel in the wind, true velocity is that compared to the ground. Therefore the speed of the wind is:

v wind = 165 - 145

v wind = 20 km/h

Therefore the wind velocity = 20 km/h against the plane.

The wind velocity affecting the plane relative to the observer is 310 km/h toward the north. This is determined by vector addition of air velocity of the plane relative to the plane and ground velocity of the plane relative to the observer.

To determine the velocity of the wind affecting the plane relative to the observer, we can use vector addition.

Given:

We need to find the wind velocity relative to the observer [tex]v_{wg}[/tex]. The relation can be expressed as:

[tex]v_{pg} = v_{ap} + v_{wg}[/tex]

Here, South and North are in opposite directions, so we can subtract these velocities and solve for vwg.

Let's assume south direction as negative and north as positive.

Calculation:

[tex]+145 km/h \text{(toward north)} = -165 km/h \text{(south)}+ v_{wg}[/tex]

Solving for [tex]v_{wg}:[/tex]

[tex]v_{wg} = 145 km/h + 165 km/h = 310 km/h[/tex]

Therefore, the velocity of the wind relative to the observer is 310 km/h toward the north.

A line that describes volume across the surface of an object or shape is called an "equidistant" line.

The line that describes volume across the surface of an object or shape is termed as an "equidistant" line.

In geometry, equidistant lines are those that have the same distance from a given point or a set of points. When considering volume across the surface of an object or shape, equidistant lines represent contours or lines of constant volume.

These equidistant lines are typically drawn parallel to each other, maintaining a consistent distance from each other across the surface of the object or shape. By connecting points on these equidistant lines, one can create contour lines or isopleths that depict variations in volume across the surface.

For example, in a topographic map, equidistant lines represent contours of constant elevation, indicating points of equal height above a reference point such as sea level.

In engineering and design, equidistant lines are essential for visualizing and understanding volume distributions within objects or shapes. They are also utilized in various fields such as geography, geology, and fluid dynamics to analyze and interpret spatial data and phenomena.

The element sulfur (S) is most likely to form covalent bonds with the oxygen element, therefore the correct answer is option D.

A chemical compound is a combination of two or more either similar or dissimilar chemical elements

for example, H₂O is a chemical compound made up of two oxygen atoms and a single hydrogen atom

As given in the problem we have to find out which of the elements sulfur (S) is most likely to form covalent bonds,

Helium is an inert gas hence it can not form a covalent bond with sulfur.

Magnesium is an electropositive element and it would form an ionic bond with the sulfur, not a covalent bond.

Thus, the element sulfur (S) is most likely to form covalent bonds with the oxygen element, therefore the correct answer is option D.

To learn more about a chemical compound, refer to the link given below ;

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According to the Gauss law, the electric flux through the closed surface is [tex]$\frac{1}{{{\varepsilon }_{0}}}$[/tex] times the charge enclosed by the surface.

[tex]$\Delta \phi =\frac{q}{{{\varepsilon }_{0}}}$[/tex]

Here, [tex]$\Delta \phi $[/tex] is the electric flux.

Gaussian surface a and b encloses the same positive point charges. So, the electric flux through surface a is four times larger than that through surface b is incorrect.

The total electric flux through surface b is eight times larger than that through surface a is incorrect because the electric flux is [tex]$\frac{1}{{{\varepsilon }_{0}}}$[/tex] times the total charge enclosed by the surface.  

As one is aware that the electric flux is independent of the area of the Gaussian surface.

The total electric flux through surface a is eight times larger than that through surface b is also incorrect because the electric flux is independent of the area of the Gaussian surface.

Explanation:

Electric flux is independent of the area of the Gaussian surface. Since the charges enclosed by the surfaces are equal, then the electric flux through the surface will be equal.

Therefore, the total electric flux through the two surfaces is equal.

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The intensity of the sound is [tex]\( 10^{-3.6} \, \text{W/m}^2 \)[/tex].

Step 1

To find the intensity (I) of a sound given its intensity level (L), we can use the formula:

[tex]\[ L = 10 \cdot \log_{10}\left(\frac{I}{I_0}\right) \][/tex]

Where:

- L is the intensity level in decibels (dB)

- I is the intensity of the sound

- [tex]\( I_0 \)[/tex] is the reference intensity (usually the threshold of hearing, which is [tex]\( 10^{-12} \) W/m²)[/tex]

Step 2

Given that the intensity level L is 84 dB and [tex]\( I_0 = 10^{-12} \)[/tex] W/m², we can rearrange the formula to solve for I:

[tex]\[ 84 = 10 \cdot \log_{10}\left(\frac{I}{10^{-12}}\right) \][/tex]

Dividing both sides by 10:

[tex]\[ 8.4 = \log_{10}\left(\frac{I}{10^{-12}}\right) \][/tex]

Step 3

Now, we can raise both sides as powers of 10:

[tex]\[ 10^{8.4} = \frac{I}{10^{-12}} \]\[ 10^{8.4} \times 10^{-12} = I \]\[ I = 10^{8.4 - 12} \]\[ I = 10^{-3.6} \, \text{W/m}^2 \][/tex]

So, the intensity of the sound is [tex]\( 10^{-3.6} \, \text{W/m}^2 \)[/tex].

Complete correct question:

What is the intensity of a sound with a measured intensity level of 84 dB?(Iσ=[tex]10^-^1^2 Watt/m^2[/tex])

The velocity of the ball at t = 1 second is 4 ft/sec.

In order to find the velocity at any given time t, we need to take the derivative of the position function y = 36t − 16t2. The derivative of this function is dy/dt = 36 - 32t. Plugging t = 1 into this derivative, we get dy/dt = 36 - 32 * 1 = 4 ft/sec. Therefore, the velocity of the ball at t = 1 sec is 4 ft/sec.

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Cars with crumple zones reduce injuries by increasing the time of impact during a collision, which decreases the forces on passengers.

A way to prevent injuries in a collision is to design cars with parts that can crumple or collapse, which help protect the passengers. The correct answer to how this helps is a. It reduces injury to the passengers by increasing the time of impact. In the event of an accident, a longer impact time means the force exerted on the car and its occupants is spread out over a longer period, resulting in less forceful impacts and thereby reducing injuries. Cars now come with features like airbags and dashboard padding which also serve to increase the time over which the force acts on occupants, reducing the forces they experience.

This question, relating to non-uniform charge distribution in a spherically symmetric manner, involves the concepts of Gaussian surfaces and charge distribution symmetry. Apply Gauss's law to solve it, we need to integrate the charge density function over the volume enclosed by the Gaussian surface. The concept of the spherical shell is crucial in the calculation of charges enclosed within the Gaussian surface.

The situation regarding the nonuniform, but spherically symmetric, distribution of charge is a problem that often comes up in physics. In this situation, there are several important concepts to understand. One of the main concepts is that of a Gaussian surface, which is an imaginary surface we define in order to apply Gauss's law.

Given that the charge density (ρ(r)) varies with r, the r here denotes the respective radius of the Gaussian surface being considered, and the use of the infinitesimal spherical shell helps facilitate calculation of the enclosed charges. To solve this, we would need to integrate the charge density function over the volume enclosed by the Gaussian surface to get the total enclosed charge.

Another key term to understand is spherical symmetry with non-uniform charge distribution. A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and does not depend on direction. Thus, even though the distribution of charges is non-uniform, as long as it only depends on the radial distance and not the direction, the charge distribution is deemed as spherically symmetrical.

When considering points outside the charge distribution, the additional volume does not contribute to the total enclosed charge, indicating the importance of the spherical shell concept that allows one to focus on the relevant range, which depends on whether the field point is inside or outside the charge distribution.

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