A gas station stores its gasoline in a tank under the ground. The tank is a cylinder lying horizontally on its side. (In other words, the tank is not standing vertically on one of its flat ends.) If the radius of the cylinder is 1.5 meters, its length is 5 meters, and its top is 5 meters under the ground, find the total amount of work needed to pump the gasoline out of the tank. (The density of gasoline is 673 kilograms per cubic meter; use g=9.8 m/s2.)

Answer:

The magnitudes of the second force is   [tex]Z = 129.9 N[/tex]

The magnitudes of the  resultant force is   [tex]R = 256.047 N[/tex]

Explanation:

From the question we are told that  

    The force is  [tex]F = 187 \ lb[/tex]

     The angle made with second force [tex]\theta_o = 73 ^o 36' = 73 + \frac{36}{60} = 73.6^o[/tex]

     The angle between the resultant force and the first force [tex]\theta _1 = 29 ^o 1 ' = 29 + \frac{1}{60} = 29.0167^o[/tex]

For us to solve problem we are going to assume that

     The magnitude of the second force is  Z N

     The magnitude of the resultant force is R N

According to Sine rule

                [tex]\frac{F}{sin (\theta _o - \theta_1 } = \frac{Z}{\theta _1}[/tex]

Substituting values

             [tex]\frac{187}{sin(73.3 - 29.01667)} =\frac{Z}{sin (29.01667)}[/tex]  

             [tex]267.82 =\frac{Z}{0.4851}[/tex]  

              [tex]Z = 129.9 N[/tex]

According to cosine rule

       [tex]R = \sqrt{F ^2 + Z^2 + 2(F) (Z) cos (\theta _o) }[/tex]

Substituting values

     [tex]R = \sqrt{187^2 + 129.9 ^2 + 2 (187 ) (129.9) cos (73.6)}[/tex]

     [tex]R = 256.047 N[/tex]

Answer:

troposphere

Explanation:

because troposphere is the layer of atmosphere closest to the earth . air is thicker at lower altitudes requiring more energy to push themselves to the sky . however , the air is thinner caused flight more fuel efficient .

Answer:

Explanation:

Given that,

Pie diameter = 9 in

Then, the circumference of the pie is

P = πd = 9π in

Then rim of the pie rotates 233 in,

Then,

1 Revolution of the pie is 9π in,

So, for 233 in, we will have

233 in / 9π in revolution

8.24 revolution

So, the revolution of the pie is 8.24

1 revolution is 2πrad

Then,

8.24 revolution = 8.24 × 2π = 51.78 rad.

And also, 1 revolution is 360°

Then,

8.24 revolution = 8.24 × 360 = 2966.4°

So,

In revolution, θ = 8.24 revolution

In radian = θ = 57.78 rad

In degree θ = 2966.4°

The angular distance should be

In revolution, θ = 8.24 revolution.

In radian = θ = 57.78 rad.

In degree θ = 2966.4°.

Since

Pie diameter = 9 in

So,  the circumference of the pie should be

P = πd = 9π in

And, rim of the pie rotates 233 in,

So,

1 Revolution of the pie is 9π in,

So, for 233 it should be

= 233 in / 9π in revolution

= 8.24 revolution

Now in the case when

1 revolution is 2πrad

So,

8.24 revolution = 8.24 × 2π = 51.78 rad.

And also, 1 revolution is 360°

So,

8.24 revolution = 8.24 × 360 = 2966.4°

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Answer:

Explanation:

Given that,

Spring constant = 16N/m

Extension of spring

x = 8cm = 0.08m

Mass

m = 5g =5/1000 = 0.005 kg

The ball will leave with a speed that makes its kinetic energy equal to the potential energy of the compressed spring.

So, Using conservation of energy

Energy in spring is converted to kinectic energy

So, Ux = K.E

Ux = ½ kx²

Then,

Ux = ½ × 16 × 0.08m²

Ux = 0.64 J

Since, K.E = Ux

K.E = 0.64 J

Answer: 10.2 kg if g = 9.8, 10 if g = 10.

Explanation:

Weight or the "force of gravity" on a person is simply defined by the equation: F = ma. In this case, the acceleration is g, which is 9.8 but can be rounded up to 10. Based on this, we have:

F = mg

100 = m*9.8

m = 10.2(or 10 if we set g to 10).

The child's mass on the earth  if the force of gravity on the child’s body is 100 N will be equal to 10.2 kg.

The fundamental force of attraction operating on all matter is recognized as gravity, also spelled gravity, in mechanics.

It has no impact on identifying the interior properties of common matter because it is the weakest force known to exist in nature.

The formation and growth of planets, galaxies, and the universe are all under the influence of this long-range, cosmic force, which further determines the trajectories of objects throughout the universe and the entire universe.

As per the given information in the question,

Weight, w = 100 N

Use the formula,

W = m × g

100 N = m × 9.8

m = 100/9.8

m = 10.2 kg

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Answer:

Dogs were probably domesticated by accident, when wolves began trailing ancient hunter-gatherers to snack on their garbage. Docile wolves may have been slipped extra food scraps, the theory goes, so they survived better, and passed on their genes. Eventually, these friendly wolves evolved into dogs

Selective breeding, also known as artificial selection, is a process used by humans to develop new organisms with desirable characteristics. Breeders select two parents that have beneficial phenotypic traits to reproduce, yielding offspring with those desired traits.

Hope it helps!

Final answer:

The speed of the 15.0-kg object as it passes the 10.0-kg object is 13.3 m/s, and the speed of the 10.0-kg object as it passes the 15.0-kg object is -13.3 m/s.

Explanation:

To determine the speeds of the two objects as they pass each other, we can use the principle of conservation of mechanical energy. When the objects are released from rest, the potential energy of the system is converted into kinetic energy as the objects fall. The sum of the kinetic energies of the two objects will be equal to the initial potential energy of the system.

Using the formula for potential energy (PE=mgh), we can calculate the initial potential energy of the system. The 15.0-kg object will fall a distance of 3.00m, so its potential energy is (15.0 kg)(9.8 m/s^2)(3.00 m) = 441 J. The 10.0-kg object will rise a distance of the same amount, so its potential energy is -441 J (we take the negative sign because the object is moving in the opposite direction).

Now, we can equate the sum of the kinetic energies of the two objects to the initial potential energy of the system. Let v1 be the speed of the 15.0-kg object and v2 be the speed of the 10.0-kg object. The kinetic energy of the 15.0-kg object is (1/2)(15.0 kg)(v1^2) and the kinetic energy of the 10.0-kg object is (1/2)(10.0 kg)(v2^2). Setting the sum of these two kinetic energies equal to 441 J, we can solve for v1 and v2.

441 J = (1/2)(15.0 kg)(v1^2) + (1/2)(10.0 kg)(v2^2)

Simplifying the equation, we have 441 J = (7.5 kg)(v1^2) + (5.0 kg)(v2^2). Since the objects are joined by a cord and the pulley does not slip, the speeds of the two objects will be equal in magnitude but opposite in direction. So we can write v2 = -v1 and substitute into the equation. We can then solve for v1:

441 J = (7.5 kg)(v1^2) + (5.0 kg)(-v1^2)

Simplifying further, 441 J = (2.5 kg)(v1^2)

Solving for v1,

v1^2 = 176.4 m^2/s^2

v1 = 13.3 m/s

Therefore, the speed of the 15.0-kg object as it passes the 10.0-kg object is 13.3 m/s, and the speed of the 10.0-kg object as it passes the 15.0-kg object is -13.3 m/s.

Answer:

The force is "19 µN".

Explanation:

The lane's j-component is meaningless, as the current is flowing in the -j line.  

Therefore the power is now in the direction of + z (out of the page if x and y are in the page plane) and has the magnitude.

[tex]\ formula: \\\\\ Forec (F) = mA \\\\ \ F \ = 2.0mA \times \int {0.3} \ y \ dy \rightarrow \ from\ 0 \ to \ 0.25 \\\\\ F \ = \ 2.0mA \times 0.15 * 0.25^{2} m\cdot T \\\\ F = 19 \µN[/tex]

Answer:

37.45 km/hr

Explanation:

To solve this, we use the law of conservation of momentum in two directions (x, and y).

in x direction

1100 * v * cosθ = 1100 * 20 * Cos30 + 1300 * 23 * cos46

1100 * vcosθ = 22000 * 0.866 + 29900 * 0.695

1100 * vcosθ = 19052 + 20780.5

1100 * vcosθ = 39832.5

vcosθ = 39832.5 / 1100

vcosθ = 36.21

In the y direction

1100 * v * Sinθ = 1100 * 20 sin30 - 1300 * 23 sin46

1100 * vsinθ = 22000 * 0.5 - 29900 * 0.719

1100 * vsinθ = 11000 - 21498.1

1100 * vsinθ = -10498.1

vsinθ = -10498.1 / 1100

vsinθ = -9.54

Since we are looking for v, then

v² = vcos²θ + vsin²θ

v² = 36.21² + (-9.54²)

v² = 1311.16 + 91.01

v² = 1402.17

v = √1402.17

v = 37.45 km/hr

Thus, the initial speed of the lighter car is 37.45 km/hr

Answer:

The right answer among the options is that: Heat is transformed, but not created or destroyed.

Explanation:

The First Law of Thermodynamics states that heat is a form of energy, and that thermodynamic processes are subject to the principle of conservation of energy.

This implies that heat energy cannot be created or destroyed. However, energy can be transferred from one location to another and converted to and from other forms of energy.

Thus, the right answer among the options is that:Heat is transformed, but not created or destroyed.

Answer:

heat is transformed, but not created or destroyed

Explanation:

Answer:

b

Explanation:

the earths mass is more than the moon .

Answer:

The object's weight would be less on the moon.

The Object's mass would be the same on the moon

Explanation:

Answer:

Find attachments for step by step solution.

A power of 68 KW is required to operate the ski lift. A power of 9660.5W is required to accelerate this ski lift.

The power can be defined as the rate of doing work, it is the work done in unit time. The SI unit of power joules per second (J/s) or Watt (W).

Power is a time-based quantity and the rate at which work is done upon an object. The formula for power can be expressed as mentioned below.

Power = Work/time

P = W/t

Given, the chairs are spaced 20 m apart a length of 1 km = 1000m

Then the number of chairs = 1000/20 = 50

Each chair weighs  = 250 kg

Then the weight of M = 50 × 250 = 12500 Kg

Consider, the initial and final heights, h₁ = 0, h₂ = 200 m

The work needed to raise the chairs, W = mgh,

W = 12500 × 9.81 × (200 - 0)

W = 2.54 × 10⁷ J

The rate of  work done at a distance of 1 km = 10 km/h,

t = 1/10 = 0.1 hr = 360 s

The power needed to operate this ski lift is, P = W/t

P = 2.54×10⁷ / 360

P = 68125 W = 68 kW

Given, the initial velocity, u = 0 m/s, final velocity, v = 10 km/h = 2.78 m/s

and, t = 5sec

Acceleration during it is first turned on is:

a = (v - u)/t

a = 2.78/ 5

a = 0.556 m/s²

The power required to accelerate this ski lift is:

P = ½ m [(v² - u²)/t]

P = ½ × 12500 × [2.78²/5]

P = 6250 × 1.55

P = 9660.5 W

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Answer:

Explanation:

energy emitted by  source per second  = .5 J

Eg = 1.43 eV .

Energy converted into radiation = .5 x .12 = .06 J

energy of one photon = 1.43 eV

= 1.43 x 1.6 x 10⁻¹⁹ J

= 2.288 x 10⁻¹⁹ J .

no of photons generated = .06 / 2.288 x 10⁻¹⁹

= 2.6223 x 10¹⁷

wavelength of photon λ = 1275 / 1.43 nm

= 891.6 nm .

momentum of photon = h / λ  ;  h is plank's constant

= 6.6 x 10⁻³⁴ / 891.6 x 10⁻⁹

= .0074 x 10⁻²⁵ J.s

Total momentum of all the photons generated

= .0074 x 10⁻²⁵  x 2.6223 x 10¹⁷

= .0194 x 10⁻⁸ Js

b ) spectral width in terms of wavelength = 30 nm

frequency width = ?

n = c / λ  , n is frequency , c is velocity of light and λ is wavelength

differentiating both sides

dn = c x dλ / λ²

given dλ = 30 nm

λ = 891.6 nm

dn = 3 x 10⁸ x 30 x 10⁻⁹ / ( 891.6  x 10⁻⁹ )²

= 11.3 x 10¹² Hz .

c )

10 nW = 10  x 10⁻⁹ W

= 10⁻⁸ W .

energy of 50 dB

50 dB = 5 B

I / I₀ = 10⁵   ;   decibel scale is logarithmic , I is energy of sound having dB = 50 and  I₀ = 10⁻¹² W /s

I = I₀ x 10⁵

= 10⁻¹² x 10⁵

= 10⁻⁷ W

= 10 x 10⁻⁸ W

power required

= 10⁻⁸ + 10 x 10⁻⁸ W

= 11  x 10⁻⁸ W.

Answer:

A) 30 s, 792 m

B) 10.28 s, 4108.2 m = 4.11 km

Explanation:

A) Traveling with an initial speed of 70 km/h, a car accelerates at 6000km/h^2 along a straight road. How long will it take to reach a speed of 120 km/h? Also, through what distance does the car travel during this time?

Using the equations of motion.

v = u + at

v = final velocity = 120 km/h

u = initial velocity = 70 km/h

a = acceleration = 6000 km/h²

t = ?

120 = 70 + 6000t

6000t = 50

t = (50/6000) = 0.0083333333 hours = 30 seconds.

Using the equations of motion further,

v² = u² + 2ax

where x = horizontal distance covered by the car during this time

120² = 70² + 2×6000×x

12000x = 120² - 70² = 9500

x = (9500/12000) = 0.79167 km = 791.67 m = 792 m

B) At t = 0 bullet A is fired vertically with an initial (muzzle) velocity of 450 m/s. When t = 3 s, bullet B is fired upward with a muzzle velocity of 600 m/s. Determine the time t, after A is fired, as to when bullet B passes bullet A. At what altitude does this occur?

Bullet A is fired upwards with velocity 450 m/s

Bullet B is fired upwards with velocity 600 m/s too

Using the equations of motion, we can obtain a relation for when vertical distance covered by the bullets and time since they were fired.

y = ut + ½at²

For the bullet A

u = initial velocity = 450 m/s

a = acceleration due to gravity = -9.8 m/s²

y = 450t - 4.9t² (eqn 1)

For the bullet B, fired 3 seconds later,

u = initial velocity = 600 m/s

a = acceleration due to gravity = -9.8 m/s²

t = T

y = 600T - 4.9T²

At the point where the two bullets pass each other, the vertical heights covered are equal

y = y

450t - 4.9t² = 600T - 4.9T²

But, note that, since T starts reading, 3 seconds after t started reading,

T = (t - 3) s

450t - 4.9t² = 600T - 4.9T²

450t - 4.9t² = 600(t-3) - 4.9(t-3)²

450t - 4.9t² = 600t - 1800 - 4.9(t² - 6t + 9)

450t - 4.9t² = 600t - 1800 - 4.9t² + 29.4t - 44.1

600t - 1800 - 4.9t² + 29.4t - 44.1 - 450t + 4.9t² = 0

179.4t - 1844.1 = 0

t = (1844.1/179.4) = 10.28 s

Putting this t into the expression for either of the two y's, we obtain the altitude at which this occurs.

y = 450t - 4.9t²

= (450×10.28) - (4.9×10.28×10.28)

= 4,108.2 m = 4.11 km

Hope this Helps!!!!

Answer:2m/s^2

Explanation:

mass=5kg

Force=10N

Acceleration=force ➗ mass

Acceleration=10 ➗ 5

Acceleration=2m/s^2

Answer:

Explanation:

The original frequency of sound being emitted f₀ = 1800

Its velocity towards the observer v ( let )

Apparent frequency f = 2130

velocity of sound = V

 [tex]f=f_0\times\frac{V}{(V - v)}[/tex]

Placing the given values

[tex]2130=1800\times\frac{343}{(343 - v)}[/tex]

1.1833 = [tex]\frac{343}{343 - v}[/tex]

1.1833 v = 62.87

v = 53.13 m /s .

b ) In the second case

formula for apparent frequency

[tex]f=f_0\times\frac{V+v}{(V - v)}[/tex]

Substituting the values

[tex]f=1800\times\frac{343+53}{(343 - 53)}[/tex]

= 2458 Hz .

Answer:

0.05m

Explanation:

Density of water = ρ(w) = 1000 kg/ m³ ;  

Density of Mercury = ρ(m) = 13628.95 kg/ m³  

Total pressure at bottom of cylinder=1.1atm

Therefore, pressure due to water and mercury =1.1-1 =atm

0.1atm=10130pa

The pressure at the bottom is given by,

ρ(w) x g[0.4 - d] + ρ(m) x g x d  = 10130

1000 x 9.8[0.4 - d] + 13628.95 x 9.8 d = 10130

3924 - 9810d + 133416d= 10130

123606d= 6206

d= 6202/123606

d= 0.05m

Depth of mercury alone =  d = 0.05m

We have that for the Question " determine the depth of the mercury."

It can be said that

From the question we are told

the cylinder is 0.4 m tall and the absolute (or total) pressure at the bottom is 1.1 atmospheres, determine the depth of the mercury. (Assume the density of mercury to be 1.36 104 kg/m^3, and the ambient atmospheric pressure to be 1.013e5 Pa)

Therefore,

Absolute pressure at the bottom of the container =

[tex]P = 1.1 atm = 1.1 * (1.013*105) Pa\\\\= 1.1143 * 10^5 Pa[/tex]

Where,

Height of the cylinder = H = [tex]0.4 m[/tex]

Height of the water in the cylinder = [tex]H_1[/tex]

Height of the mercury in the cylinder = [tex]H_2[/tex]

Therefore,[tex]H = H_1 + H_2\\\\H_1 = H - H_2\\\\P = P_{atm} + \rho_1gH_1 + \rho_2gH_2\\\\P = P_{atm} + \rho_1g(H - H_2) + \rho_2gH_2\\\\1.1143*10^5 = 1.013*10^5 + (1000)(9.81)(0.4 - H_2) + (1.36*10^4)(9.81)H_2\\\\1.013*10^4 = 3924 - 9810H_2 + 133416H_2\\\\143226H_2 = 6206\\\\H_2 = 4.333*10^{-2} m[/tex]

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Answer:

[tex]L(t) = 0.125 (cos (\frac{\pi}{14.765}(t+7)) + 0.125[/tex]

Explanation:

The expression for the  trigonometric function is :

L(t) = A (cos (B(t - C)))+ D   ----- equation (1)

where ;

[tex]A = \frac{max-min}{2}[/tex]

[tex]A = \frac{0.25-0}{2}[/tex]

A = 0.125

D = [tex]\frac{0+.025}{2}[/tex]

D = 0.125

Period of the lunar cycle = 29.53

Then;

[tex]\frac{2 \pi}{B} = 29.53[/tex]

[tex]29.53 \ \ B = 2 \pi[/tex]

[tex]B = \frac{2 \pi}{29.53}[/tex]

[tex]B = \frac{\pi}{29.53}[/tex]

Also; we known that December 25 is 7 days before January 1.

Then L(-7) = 0.025

Plugging all the values into trigonometric function ; we have:

[tex]0.125 ( cos ( \frac{\pi}{14.765}((-7)-C)))+0.125 = 0.25 \\ \\ \\ ( cos ( \frac{\pi}{14.765}((-7)-C))) = \frac{0.25-0.125}{0.125}[/tex]

[tex]( cos ( \frac{\pi}{14.765}((-7)-C))) = 1[/tex]

[tex]( \frac{\pi}{14.765}((-7)-C))= cos^{-1} (1)[/tex]

[tex]}((-7)-C))=0[/tex]

[tex]C= -7[/tex]

[tex]L(t) = 0.125 (cos (\frac{\pi}{14.765}(t-(-7))) + 0.125[/tex]

[tex]L(t) = 0.125 (cos (\frac{\pi}{14.765}(t+7)) + 0.125[/tex]

The formula for the trigonometric function that models the illumination [tex]\( L(t) \)[/tex] of the moon t days after January 1, 2016, is:

[tex]\[ L(t) = 0.125 + 0.125 \cos\left(\frac{2\pi}{29.53}(t - 777 - 25)\right) \][/tex]

The moon's illumination cycle can be modeled using a cosine function because it is periodic and symmetric about the maximum (full moon) and minimum (new moon) points. The maximum illumination is 0.250 lux, and the minimum is 0 lux. The period of the cycle is 29.53 days.

To find the midline of the function, we take the average of the maximum and minimum illuminations:

[tex]\[ \text{Midline} = \frac{\text{Max illumination} + \text{Min illumination}}{2} = \frac{0.250 + 0}{2} = 0.125 \][/tex]

The amplitude of the function is the distance from the midline to the maximum or minimum, which is half the difference between the maximum and minimum illuminations:

[tex]\[ \text{Amplitude} = \frac{\text{Max illumination} - \text{Min illumination}}{2} = \frac{0.250 - 0}{2} = 0.125 \][/tex]

The period of the function is the length of one complete cycle, which is given as 29.53 days. To convert this to radians, we use the formula:

[tex]\[ \text{Radian frequency} = \frac{2\pi}{\text{Period}} = \frac{2\pi}{29.53} \][/tex]

The phase shift occurs because the full moon does not happen on January 1, 2016. It happens on December 25, 2015, which is 777 days before January 1, 2016, plus an additional 25 days to account for the days after December 25 until January 1. Therefore, the phase shift is:

[tex]\[ \text{Phase shift} = 777 + 25 \][/tex]

Putting all this together, the formula for the illumination L(t) as a function of time t (in days) is:

[tex]\[ L(t) = \text{Midline} + \text{Amplitude} \cdot \cos\left(\text{Radian frequency} \cdot (t - \text{Phase shift})\right) \]\[ L(t) = 0.125 + 0.125 \cos\left(\frac{2\pi}{29.53}(t - 777 - 25)\right) \][/tex]

This function models the illumination of the moon t days after January 1, 2016, with the correct midline, amplitude, period, and phase shift.

Answer:

If Patty remains on Earth then at the time Peggy will come back from her trip, Peggy will be much younger than her sister Patty because of time-dilation.

Explanation:

Peggy and Patty are sisters. Peggy is an astronaut and Patty is an astronomer.

Peggy goes for mission to Alpha Centauri, the nearest star system to the Solar System at 80% of the speed of light, and will come back after 1010 years.

If Patty remains on Earth then at the time Peggy will come back from her trip, Peggy will be much younger than her sister Patty because of time dilation.

This is due to the fact that time moves slower in Alpha Centauri because of its massive gravitational force which bends space time. Moreover, It is known that Peggy's spacecraft moves at 80% of the speed of light, it will result in velocity time dilation since time moves slow if you travel at a speed near to the speed of light.

Final answer:

Peggy, the astronaut twin traveling at 80% of the speed of light, will experience less time due to time dilation, and upon her return will be younger than her Earth-bound twin sister, Patty.

Explanation:

The question is about the relativistic effects that occur when one twin travels at significant fraction of the speed of light while the other remains on Earth.

According to the theory of relativity, time dilation will cause the traveling twin, Peggy, to age more slowly compared to her twin sister, Patty, who remains on Earth.

If Peggy travels to Alpha Centauri, which is 4.3 light years away, at 80% of the speed of light, and assuming the round trip takes 10 years for the Earth-bound twin, we can calculate that the moving twin will experience less than 10 years of elapsed time due to the effects of time dilation.

This happens because the faster Peggy travels, the more pronounced the effect of time dilation will be. This is a well-known result predicted by Einstein's special theory of relativity and has been confirmed through experiments involving high-speed particles and precise clocks.

Thus, when Peggy returns, she will be younger than her twin sister Patty, who has experienced the full 10 years on Earth.

Answer:

Explanation:

Mass of satellite

M_s = 361 kg

Distance of satellite from moon

h = 147 km = 147,000m

Radius of the moon is

R_m = 1740 km = 1740,000m

Mass of the moon is

M_m = 7.36 × 10²² kg.

The kinetic energy is equal to the potential energy of the body to the surface of the moon from the conservation of energy.

K.E = P.E = mgh

Gravity on moon is g = 1.62 m/s²

K.E = 361 × 1.62 × 147,000

K.E = 8.597 × 10^7 J.

B. The gravitational potential energy can be calculated using

U = G•M_s × M_m (1/R_s - 1 / R)

R is the total distance from the centre of the moon to the satellite

R = h + R_m = 147 + 1740 = 1887km

R = 1,887,000 m

U = 6.67 × 10^-11 × 361 × 7.36 × 10²² (1/1,740,000 - 1/1,887,000)

U = 6.67 × 10^-11 × 361 × 7.36 × 10²² × 4.48 × 10^-8

U = 7.93 × 10^7 J

Then,

The total energy becomes

E = K.E + U

E= 8.597 × 10^7 + 7.93 × 10^7 J

E = 1.653 × 10^8 J

Answer:

Explanation:

y = y₀ cos[tex]\sqrt{\frac{k}{m} }\times t[/tex]

a )

[tex]\frac{dy}{dt}[/tex] = - y₀ x [tex]\sqrt{\frac{k}{m} }[/tex] sin (  [tex]\sqrt{\frac{k}{m} }\times t[/tex]  )  

b ) If m = 4m

[tex]\frac{dy}{dt}[/tex] = - y [tex]\sqrt{\frac{k}{4m} }[/tex] sin ( [tex]\sqrt{\frac{k}{4m} }\times t[/tex]  )

Magnitude of velocity will be decreased .

c )

[tex]\frac{dy}{dt}[/tex] = - y [tex]\sqrt{\frac{4k}{m} }[/tex] sin ( [tex]\sqrt{\frac{4k}{m} }\times t[/tex]  )

magnitude of velocity will be increased .

d )

velocity = - y₀ [tex]\sqrt{\frac{k}{m} }[/tex] sin( [tex]\sqrt{\frac{k}{m} }\times t[/tex]  )

              =  L [tex]\sqrt{\frac{ms^{-2}}{m} }[/tex]  X 0

               = L s⁻¹

= m /s

unit of velocity is consistent .

Final answer:

The velocity of a mass attached to a spring is found by differentiating the position equation with respect to time, revealing how the system's velocity changes with variations in mass and spring stiffness. Increasing the mass decreases the velocity amplitude, while increasing the spring stiffness increases it.

The units of velocity, m/s, are confirmed through dimensional analysis.

Explanation:

To find the velocity of the mass, we need to differentiate the position equation y = yocos (t √ k/m) with respect to time. Using the chain rule for differentiation, the derivative of y with respect to t gives us dy/dt = -y_0(√ k/m)sin(t √ k/m), where dy/dt represents the velocity of the mass.

This equation tells us the velocity at any given moment for a mass m and spring constant k.

b) If the mass is increased by four times, the equation for velocity becomes dy/dt = -y_0(√ k/(4m))sin(t √ k/(4m)). The increase in mass causes the velocity amplitude to decrease, as √(1/4) is in the equation, indicating that velocity decreases in proportion to the square root of the mass increase.

c) Increasing the spring constant k fourfold results in the new velocity equation dy/dt = -y_0(√ (4k)/m)sin(t √ (4k)/m). This shows an increase in the velocity amplitude, as the increase in k results in a velocity proportional to the square root of the increase in k, thus making the system oscillate faster.

d) To confirm the units of velocity are meters per second (m/s), we substitute the units into the derivative of the position equation: [m]*[s√(kg/s2)/kg] simplifies to m/s, thus showing the units of velocity are indeed consistent and correct.

Answer:

9.6 ft

Explanation:

Distance is inversely proportional to weight

distance = k / (weight), where

k is a constant

or you could say,

distance * weight = k

In this scenario,

120 * 16 = 200 * distance

On rearranging, making, distance the subject of formula, we have

Distance = 120 * 16 / 200

Distance = 1920 / 200

Distance = 9.6 ft

So the 200 pounds person should sit 9.6 feet away from the centre to balance the see saw

Answer:

4.8 ft

Explanation:

torque = wt × distance

t1 = 120lb x 8 ft =960

t2 = 200lb x X ft

set them equal to each other.

120(8) = 200x

960 = 200x

x = 4.8 ft

At 0.108 m distance from the center of the sphere does the electric field have the same magnitude as it has at P.

Given that,

Radius of the sphere R= 0.30 m

Distance from the center of the sphere r= 0.50 m

Electric field = 15000 N/C

r > R and for this value of r.

Let density of charge = [tex]\rho[/tex]

therefore, we have,

[tex]k*\rho*(4/3*(\pi*0.3^3))/0.5^2 = k*\rho*(4/3*(\pi*r^3))/r^2[/tex]

where r is the distance from centre.

Now, we have,

the field inside a sphere is given by [tex]\(kqx/r^3\)[/tex]

the field outside the sphere is given by [tex]\(kq/x^2\)[/tex]

so equating the two equations,

[tex]kq*x/(0.3^3)=kq/(0.5^2)[/tex]

or,  x=0.108m

So, After solving we get distance= 0.108 m.

Hence, At 0.108 m distance from the center of the sphere does the electric field have the same magnitude as it has at P.

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The electric field of a nonconducting sphere is directly proportional to the distance from its center when inside the sphere and inversely proportional to the square of the distance when outside. To have the same magnitude of 15,000 N/C as at point P (0.50m), the point must be outside the sphere at a distance that depends on the sphere's total charge.

The question asked is based on the concept of electric fields in Physics. Firstly, we need to understand that the electric field inside a uniformly charged nonconducting sphere is directly proportional to the distance from the center of the sphere (It follows the equation E = k*r, where E is the electric field, k is a constant, and r is the distance from the center). At the point P (0.50m), the electric field E is given as 15,000 N/C.

So, for the electric field to have the same magnitude at another point, this point must be outside the sphere. This is because the electric field will decrease once we move out of the sphere (Outside the sphere, the electric field falls off as 1/r^2, so to achieve the same magnitude of 15,000 N/C we have to move farther away from the sphere). The exact distance depends on the total charge of the sphere, which is not given in the question.

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Answer:

The current is 0.148 amps

Explanation:

To find the current, you divide the voltage given by the resistence:

8V ÷ 54 ohms

Which equals 0.1481481481 amps

The current resistor is  0.148 amps.

Ohm's law states that the voltage across a conductor is directly proportional to the current flowing through it, provided all physical conditions and temperatures remain constant.

Ohm's Law formulas V = IR, I = V/R, and R= V/I.

Current through each resistor can be found using Ohm's law I=V/R, where the voltage is constant across each resistor.

The given values are:

V = 8V and R = 54Ω

By using the formula :-

I=V/R

I =8V / 54Ω = 0.148 amps

Therefore,

0.148 amps is the current resistor.

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Answer:

The wavelength of the radiation absorbed by ozone is 319.83 nm

Explanation:

Given;

frequency of absorbed ultraviolet (UV) radiation, f = 9.38×10¹⁴ Hz

speed of the absorbed ultraviolet (UV) radiation, equals speed of light, v = 3 x 10⁸ m/s

wavelength of the absorbed ultraviolet (UV) radiation, λ = ?

Apply wave equation for speed, frequency and wavelength;

v = fλ

λ = v / f

λ = (3 x 10⁸) / (9.38×10¹⁴)

λ = 3.1983 x 10⁻⁷ m

λ = 319.83 x  10⁻⁹ m

λ = 319.83 nm

Therefore, the wavelength of the radiation absorbed by ozone is 319.83 nm

Answer:

0.52rad/s^2

Explanation:

To find the angular acceleration you use the following formula:

[tex]\omega^2=\omega_o^2+2\alpha\theta[/tex]   (1)

w: final angular velocity

wo: initial angular velocity

θ: revolutions

α: angular acceleration

you replace the values of the parameters in (1) and calculate α:

[tex]\alpha=\frac{\omega^2-\omega_o^2}{2\theta}[/tex]

you use that θ=22 rev = 22(2π) = 44π

[tex]\alpha=\frac{(12rad/s)^2-(0rad/s)^2}{2(44\pi)}=0.52\frac{rad}{s^2}[/tex]

hence, the angular acceñeration is 0.52rad/s^2

Answer:

Explanation:

The magnitude is 0.5 N

The direction is 0°

Answer:

Magnitude: 0.5 N

Direction: 0 degrees

Explanation:

Going to be honest, I do not understand the magnitude of R, but what I can explain is the direction.

So, following the steps, we find that the magnitude of A (coulomb's constant k times (q1 x q3)/dq13^2) = 1.798 Newtons

The magnitude of B is the same because q1 = q2 in this scenario

Using SohCahToa, we can find the angle of vector A = about 36.9 degrees

since q1 is 0.3 meters away from the x-axis (this is our opposite side) and the distance between q1 and q3 is 0.5 (our hypotenuse because the x-axis and y-axis make a right angle) this means that we can use this for our formula,

the sine of angle A = opposite side / hypotenuse side

Sine of angle A = 0.3m / 0.5m

Angle A = (inverse Sine) of 0.3m/0.5m

Angle A = 36.9 degrees

Using that we can get the x and y components of each vector:

The x component of A will be A cosine angle A which is

Ax = 1.798 N times (cosine(36.9 degrees)) = 1.44 N

The y component of A will be NEGATIVE A sine angle A

The reason why it is negative is because for the A vector, it has a negative slope so its y value is continually decreasing, so it makes sense for it's resulting vector to have the same y decrease.

Ay = (-1.798)(sine(36.9)) = -1.079 N

Now we can do the same with B.

Since B = A IN THIS SCENARIO, the values will be the same,

Bx = 1.798 N times (cosine(36.9 degrees) = 1.44 N

BUT with the y value here, it is actually INCREASING

so:

By = 1.798 N times (sine(36.9 degrees)) = 1.079 N

So the next step is to sum these values.

Rx = Ax + Bx = 1.44 N + 1.44 N = 2.88 N

The x component of vector R is 2.88 N.

Ry = Ay + By = -1.079 N + 1.079 N = 0 N.

The y component of vector R is 0.

Basically, it is traveling straight along the x-axis. This makes sense because the forces of each particle were repelling this one at an EQUAL MAGNITUDE (q1 = q2) So, the direction is 0 degrees because even following the steps, (tan-1 times (Ry over Rx)) is 0/2.88. This will be 0.

Answer:

1. If turning off one device turns off all the devices then it means home is wired in series.

2. Headlights of a car are connected in parallel.

3. Fuse is installed in series with a device.

4. The device will get damaged to excessive current

5. Refer to the explanation

Explanation:

1. How do you know that homes are not wired in series?

We use parallel wiring in home wiring so that each device gets the same voltage, if we use a series connection then each device will get different voltage depending upon its resistance. Moreover, in case you want to use more than 1 device then you would have to turn on both of them to complete the circuit, these are the reasons we dont use series wiring at homes.

2. Are the two headlights of a car in series or parallel?

The headlights of a car are wired in parallel so that even if one headlight gets damaged and stops working, the other headlight keeps on working. If it was wired in series then both would have stopped working when any of them gets damaged.

3. Are fuses put in parallel or in series with devices they are meant to protect?

Fuses are always connected in series, when a high fault current flows through the fuse, it gets melted and breaks the path so that the fault current doesn't flow through the device.

4. Why is it unsafe to replace a fuse that keeps burning out with one which has a higher amp rating?

The ampere rating of the fuse is selected with respect to the device it is connected to. Lets say you have a device which cannot withstand a current of 10A. So you have connected a fuse of 10A rating in series with the device. The fuse got burnt out several times now if you decided to replace the fuse with a higher ampere rating lets say 12A then what would happen? it means that now a current of 10A can flow through the device which will damage the device for sure.

5. A bulb and pair of batteries might have the batteries connected to each other in series or parallel. What are the advantages of each arrangement?

Advantages of series arrangement:

Advantages of parallel arrangement:

The frequency of the third normal mode of a guitar string with a length of 40.0 cm, mass of 0.5 g when stretched with a tension of 80N is approximately 948.68 Hz.

To calculate the frequency of the third normal mode (
n = 3) for a guitar string, we can use the formula for the frequency of a string fixed at both ends:

f_n = (n/2L) √(T/μ)

where:

f_n is the frequency of the nth mode,

n is the mode number (which is 3 in this case),

L is the length of the string,

μ is the mass per unit length of the string (linear mass density), and

T is the tension in the string.

Given the length of the string L = 40.0 cm = 0.4 m, the mass m = 0.5 g = 0.0005 kg, and the tension T = 80 N, we first need to calculate the linear mass density:

μ = m/L

In this case,

μ = 0.0005 kg / 0.4 m = 0.00125 kg/m

Now, we use the frequency formula to find f_3:

f_3 = (3/2 0.4 m √(80 N/0.00125 kg/m)
= (3/(0.8 m)) √(64000 N/m)
= 3.75 √(64000 N/m)
= 3.75 √(64000 N/m)
= 3.75 * 252.9822 Hz
= 948.68 Hz

The frequency of the third normal mode of the guitar string is approximately 948.68 Hz.