Consider the vibrations of a 2-dimensional square lattice with lattice constant a. There is only one atom in each primitive cell. Atoms have a mass M, and they can only oscillate out of the plane (transverse motion). We assume only nearest neighbour interactions (spring between nearest neighbour atoms) with a force constant a. Let Um denote the transverse displacement from equilibrium of the atom in the / row and the m column of the 2D lattice. (a) Show that the equation of motion for the displacements in is: M am = a[(u1+1,m + U1–1,m - 2uim) + (U1,m+1 + U1m-1 - 2uim)] (b) Show that the dispersion relation is given by the expression Mw2 = 2a[2 - cos(qxa) - cos(qya)]

The work done delivered by the Carnot engine is 93.75 J.

Solution:

\(T_c = 260 K\\ T_h = 320K\\ Q_h = 500J\)

As we know that Carnot engine is ideal, the efficiency depends only on the temperature of the reservoirs(in kelvin scale):

\(n = 1 - \frac{T_c}{T_h} \\n = 1 - \frac{260}{320}\)

\(n = 1- 0.8125\\n = 0.1875\)

Also, we know that the work done is equal to product of efficiency and heat at hot reservoir:

\(W = n\times Q_h\\W = 0.1875 \times 500\\W = 93.75 J\)

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Answer and Explanation: The geosphere interacts with the cryosphere when glaciers and sheets of ice from the cryosphere erode the rocks located on the geosphere. This happens as ice passes over the land and carries eroded rocks to new locations. The cryosphere can also cause soil to become icy and take hold of greenhouse gases from the atmosphere. The geosphere is the part of the Earth that has rocks, minerals, sand, and lava. The cryosphere has frozen water, ice, and glaciers. The cryosphere also interacts frequently with the hydrosphere because it draws water from the hydrosphere and freezes it during cold temperatures. Warmer temperatures cause the cryosphere to melt and release water back to the hydrosphere.

The type of intelligence required by a profession depends on its job description. Lawyers most likes have analytical intelligence.

A renowned psychologist named Howard Gardner devised a theory of multiple intelligence. This means that nobody is really not intelligent because a person must posses one or more of these multiple intelligence.

A lawyer has a lot to do with analytic reasoning and comparison of evidence to determine their validity. This task requires a strong analytical mind hence a lawyer requires analytical intelligence.

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The unknown wavelength of light is approximately 633 nm.

When light passes through a pair of slits, it creates an interference pattern consisting of dark and bright fringes. The position of these fringes depends on the wavelength of light and the distance between the slits. In this scenario, we have two wavelengths of light: 700 nm and an unknown wavelength.

The given information states that the 10th bright fringe of the unknown wavelength overlaps with the 9th bright fringe of the 700 nm light. This overlapping occurs when the path difference between the two wavelengths is equal to the wavelength of either of the lights. Since the 10th bright fringe of the unknown wavelength overlaps the 9th bright fringe of the 700 nm light, it implies that the path difference for the unknown wavelength is equal to the wavelength of the 700 nm light.

To find the unknown wavelength, we can use the formula for path difference in the double-slit interference pattern: Δx = λ * d / D, where Δx is the path difference, λ is the wavelength, d is the distance between the slits, and D is the distance from the slits to the screen.

Since the path difference for the unknown wavelength is equal to the wavelength of the 700 nm light, we can set up the following equation: (10λ_unknown) = (9λ_700). Solving for λ_unknown, we get λ_unknown ≈ (9/10) * λ_700 ≈ (9/10) * 700 nm ≈ 630 nm.

Therefore, the unknown wavelength of light is approximately 633 nm.

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Answer:

9.8 meters per square second

Explanation:

Free Falling Object. the value of g is 9.8 meters per square second on the surface of the earth. The gravitational acceleration g decreases with the square of the distance from the center of the earth. But for many practical problems, we can assume this factor to be a constant.

In order to describe the dimensional formula of this coefficient, first let's write the formula for the friction force:

Where Ff is the friction force (in Newtons), μ is the friction coefficient and N is the normal force (in Newtons).

Changing each variable to its corresponding unit (except for the coefficient), we have:

Solving for μ, we have:

Since μ = 1, that means it has no dimensions, it's only a constant value (it is an adimensional value).

The average person's vertical jump on the Moon would be 0.65 m.

The gravitational acceleration near the surface of the Moon is 1.62 m/s2, which is about one sixth the gravitational acceleration on Earth.

As a result, an average person's vertical jump on the Moon would be less than on Earth.

To calculate the vertical jump on the Moon, we need to use the formula h = 1/2 x g x t2.

This equation is used to calculate the height h (in meters) that an object will reach when thrown into the air, given the gravitational acceleration g (in m/s2) and the time t (in seconds) it takes to reach the peak of the jump.

Since the gravitational acceleration on the Moon is 1.62 m/s2, and the time taken to reach the peak of the jump is the same (assume 0.5 s), then h = 0.5 x 1.62 x (0.5)2, which is 0.65 m.

Therefore, an average person's vertical jump on the Moon would be 0.65 m.

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Answer:

Wave having a length of 9 meters and a height of 6 meters.

Explanation:

Assuming that this is a sound wave or an electromagnetic wave, we know that those have a constant speed.

The speed of sound and speed of light are the same for any sound or electromagnetic wave respectively.

And we also have the equation:

v = λ*f

Where:

v = velocity

f = frequency

λ = wavelength

And because v is a constant, we can write:

f = v/λ

Now the wavelength is in the denominator, then the wave with the smaller wavelength will be the one with a larger frequency.

The option with the smallest wavelength is

Wave having a length of 9 meters and a height of 6 meters.

Notice that the height of the waves is also given, this is actually related to the intensity of the wave, so that information does not affect this particular question.

A) The change in speed is 0.283 m/s in the opposite direction. B) The force exerted is 817.3077 Newtons. C) The kinetic energy is 557.6 Joules. D) The kinetic energy is 60.1165 Joules.

PART A:

To find the change in the speed of the space capsule, we can apply the law of conservation of momentum. The initial momentum of the astronaut-capsule system is zero since they are at rest.

After the astronaut pushes off, the total momentum remains constant. The momentum of the astronaut is given by:

P_astronaut = mass_astronaut * velocity_astronaut = 160 kg * 2.65 m/s

According to the law of conservation of momentum, the momentum of the capsule is equal in magnitude but opposite in direction to the momentum of the astronaut. So, the momentum of the capsule is:

P_capsule = -P_astronaut = -160 kg * 2.65 m/s

The change in speed of the space capsule is the difference between its final speed (which we'll call v_final) and its initial speed (which is zero):

Change in speed = v_final - 0 = v_final

Therefore, the change in speed of the space capsule is equal to the magnitude of the momentum of the astronaut divided by the mass of the capsule:

Change in speed = |P_capsule| / mass_capsule = (160 kg * 2.65 m/s) / 1500 kg

PART B:

To find the average force exerted by each one on the other, we can use Newton's second law of motion, which states that force is equal to the rate of change of momentum.

The average force exerted by the astronaut on the capsule (F_astronaut) and the average force exerted by the capsule on the astronaut (F_capsule) is equal in magnitude but opposite in direction.

Using the given time interval (t = 0.520 s), we can calculate the average force exerted:

F_astronaut = (P_capsule - P_capsule_initial) / t

F_capsule = (P_astronaut - P_astronaut_initial) / t

Since the initial momenta of the astronaut and the capsule are zero, the equations simplify to:

F_astronaut = P_capsule / t

F_capsule = P_astronaut / t

PART C:

The kinetic energy of an object can be calculated using the formula:

Kinetic energy = (1/2) * Mass * (Velocity)^2

For the astronaut, the mass is given as 160 kg, and the velocity after the push is 2.65 m/s. Substituting these values into the formula:

The kinetic energy of the astronaut = (1/2) * 160 kg * (2.65 m/s)^2

The kinetic energy of the astronaut ≈ 557.2 Joules

Therefore, the kinetic energy of the astronaut after the push is approximately 557.2 Joules.

PART D:

The kinetic energy of the space capsule can be calculated using the same formula as in Part C. The mass of the space capsule is given as 1500 kg, and the final velocity after the push is 0.283 m/s.

The kinetic energy of the space capsule = (1/2) * 1500 kg * (0.283 m/s)^2

The kinetic energy of the space capsule ≈ 60.28 Joules

By plugging in the appropriate values into the equations, the change in speed of the space capsule, the average force exerted by each on the other, the kinetic energy of the astronaut after the push, and the kinetic energy of the space capsule after the push can be calculated accurately.

A) The change in speed of the space capsule is 0.283 m/s in the opposite direction.

B) The average force exerted by each on the other is 817.3077 Newtons.

C) The kinetic energy of the astronaut after the push is 557.6 Joules.

D) The kinetic energy of the space capsule after the push is 60.1165 Joules.

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Rutherford  conducted the experiment that showed that atoms were mostly made up of empty space.

Rutherford done an experiment in which he allowed alpha particle to bombard on gold foil . Which shows  that the atom is mostly empty space with a tiny, dense, positively-charged nucleus. Based on these results, Rutherford proposed the nuclear model of the atom.

The observations made by Rutherford led him to conclude that: A major fraction of the α-particles bombarded towards the gold sheet passed through the sheet without any deflection, and hence most of the space in an atom is empty.

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The mass of the other star in the binary system is approximately 0.541 solar masses.

To find the mass of the other star in the binary system, we can use Kepler's Third Law of Planetary Motion, which can be applied to binary star systems. The law states that the square of the orbital period (\(T\)) is proportional to the cube of the semi-major axis (\(a\)) of the orbit. Mathematically, this can be expressed as\(\(T^2 = \frac{4\pi^2}{G(M_1 + M_2)}a^3\), where \(M_1\) and \(M_2\)\)  are the masses of the stars,\(\(G\)\) is the gravitational constant, and other variables have their usual meanings.

Given that one star has a mass of 1.00 solar masses, we can substitute the known values into the equation and solve for\(\(M_2\)\). Rearranging the equation, we have\(\(M_2 = \frac{4\pi^2}{G}(\frac{a^3}{T^2}) - M_1\)\).

Plugging in the values for\(\(a\) (1.34E+10 km) and \(T\) (400 years)\), and using the appropriate unit conversions, we can calculate the mass of the other star,\(\(M_2\\), to be approximately 0.541 solar masses.

Therefore, the mass of the other star in the binary system is approximately 0.541 solar masses.

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The electric field at the center of a square of side 2.0 cm is 0, and the potential at the center is also 0.

The electric field at the centre of a square an be identified by

1. In a square, the electric field at the center is zero due to the symmetry of the charges.

2. Since the square is neutral and has equal amounts of positive and negative charges distributed symmetrically, the net electric field at the center cancels out to zero.

3. The potential at a point is the amount of work done to bring a unit positive charge from infinity to that point.

4. In this case, since the square is neutral and the potential is a scalar quantity, the potential at the center is also zero.

5. Therefore, both the electric field and potential at the center of the square of side 2.0 cm are zero.

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Answer:

The average speed of the cyclist is 17.14 km/hr

Explanation:

Given;

total distance traveled by the cyclist's, d = 60 km

time taken by the cyclist, t = 3.5 hours

The average speed of the cyclist is given by;

average speed  = total distance traveled / total time taken

average speed = 60 km / 3.5 hr

average speed  = 17.14 km/hr

Therefore, the average speed of the cyclist is 17.14 km/hr

Interferometry uses two or more telescopes to achieve higher resolution than a single telescope.

When light from two or more telescopes is combined, the waves interfere with each other.

This interference can be constructive, in which the waves add together to create a brighter signal, or destructive, in which the waves cancel each other out.

The pattern of constructive and destructive interference creates an interference pattern, which can be used to measure the size and shape of objects.

The resolution of an interferometer is determined by the distance between the telescopes.

The larger the distance, the higher the resolution. This is because the interference pattern is spread out over a larger area, which allows for more detail to be seen.

Interferometry is a powerful tool for astronomers, as it allows them to see objects that would be invisible to a single telescope. It is also used in other fields, such as medical imaging and engineering.

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Interferometry uses two or more telescopes combined to produce a resolution similar to that of a larger telescope. The resolution depends not on the size of the telescopes but on their separation. Advancements in technology and computing power have made it possible to separate larger interferometer arrays for greater resolution.

Interferometry uses two or more telescopes in an interferometer array to achieve a resolution equivalent to that of a single telescope with a diameter equal to the baseline separating the individual telescopes. Instead of interfering with each other, the telescopes in the interferometer array work cooperatively, utilising the technique of interference to achieve a superior resolution.

An interferometer combines electromagnetic radiation from two or more telescopes and the resolution depends not on their individual apertures but on their separation. As such, two telescopes separated by 1 kilometre, would provide the same resolution as a single dish 1 kilometre across, although they wouldn’t collect as much radiation.

Earlier, the size of interferometer arrays used to be limited because all the dishes needed to be physically connected. However, with current technology and increased computing power, larger interferometer separations can be achieved without requiring a physical connection between telescopes. The precise arrival of electromagnetic waves from space at each telescope can be timed and this data is later combined to yield an extremely high-resolution image.

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Main answer: The magnetic field between the hot and neutral wires supplying DC power to the light rail commuter train is 0.0053 T.

The magnetic field between two parallel conductors can be calculated using the equation B = (μ0*I)/(2π*r), where B is the magnetic field, μ0 is the permeability of free space, I is the current, and r is the distance between the wires. Plugging in the given values, we get:

B = (4π x 10^-7 T*m/A)*(800 A)/(2π*0.75 m)

B = 0.0053 T

Therefore, the magnetic field between the hot and neutral wires supplying DC power to the light rail commuter train is 0.0053 T.

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Answer:

Answer:- Volume of the balloon is 5.78 L.

Solution:- There are 0.24 moles of hydrogen gas in a balloon at 35 degree C and 1.05 atm pressure. It asks to calculate the volume of the balloon.

This problem is based on ideal gas law equation:

P = 1.05 atm, n = 0.24 mole, T = 35 + 273 = 308 K

R =

V = ?

The equation could be rearranged for the volume as:

Let's plug in the values and do the calculations to get the volume of the balloon:

V = 5.78 L

So, the volume of the gas balloon is 5.78 L.

a) The correct average velocity of the man is approximately 0.026928 km/h.

b) The correct average speed of the man is approximately 0.03726 km/h.

a) Average Velocity:

To calculate the average velocity, we need to determine the total displacement and divide it by the total time taken.

The north displacement is 28.125 m and the east displacement is 18.45 m. Since we are given that each block represents 1 km, we need to convert these distances to kilometers.

North displacement: 28.125 m ÷ 1000 = 0.028125 km

East displacement: 18.45 m ÷ 1000 = 0.01845 km

The total displacement can be calculated using the Pythagorean theorem:

Total displacement = √((0.028125 km)² + (0.01845 km)²) = √(0.000790039 km² + 0.0003409025 km²) ≈ √0.001130941 km² ≈ 0.03366 km

Since the total time taken is given as 1.25 hours, the average velocity is calculated by dividing the total displacement by the total time:

Average velocity = Total displacement ÷ Total time = 0.03366 km ÷ 1.25 hours ≈ 0.026928 km/h

Therefore, the correct average velocity of the man is approximately 0.026928 km/h.

b) Average Speed:

To calculate the average speed, we need to determine the total distance traveled and divide it by the total time taken.

The total distance traveled is obtained by summing up the north and east distances:

Total distance = 0.028125 km + 0.01845 km = 0.046575 km

Average speed = Total distance ÷ Total time = 0.046575 km ÷ 1.25 hours ≈ 0.03726 km/h

Therefore, the correct average speed of the man is approximately 0.03726 km/h.

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No two electrons in an atom can have identical four quantum numbers. This is a statement that is attributed to the Pauli Exclusion Principle.

It states that in an atom or molecule, no two electrons can possess the same four electronic quantum numbers. As an orbital can accommodate a maximum of only two electrons, the two electrons must have opposite spins. This means if one electron is allotted a spin-up electron, the other electron must be a spin-down electron.

The adequacy of an orbital to accommodate electrons is decided by Pauli's exclusion principle. According to this principle, for an electron belonging to the same orbital, the spin quantum number must be different since the other three quantum numbers are the same.

The spin quantum number can have two values: +1/2 and -1/2.

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Answer:

It can be measured by using laboratory thermometer.

The ratio of angular velocity of the ball at end of the plane to its angular velocity (passing the center point C of the inclined plane) will be equivalent to "sqrt (5/2)."

The principle of conservation of energy can be applied here to determine the required ratio. As the ball rolls down the inclined plane, the potential energy it possessed at the top is transformed into kinetic energy as well as rotational energy.

Now, let's assume the ball has a mass 'm', a radius 'r', and an inertial moment of '\(I = (2/5)mr^{2}\)' (for a solid sphere). Let 'h' represent the height of the inclined plane's top and 'θ' represent the angle that the plane makes with the horizontal.

By applying the principle of conservation of energy, we get:

\(mgh = (1/2)I\omega^2 + (1/2)mv^2\)

where, \(mgh\) = potential energy of the ball at the top of the plane ('g' denoting acceleration due to gravity), \((1/2)I\omega^2\) = rotational kinetic energy ('ω' being the angular velocity), and \((1/2)mv^2\) denoting the translational kinetic energy ('v' is the linear velocity) of the ball.

On simplifying the equation, we get:

\(\omega f/\omega c = \sqrt{5r/h}\)

where \(\omega f\)= angular velocity of the ball at end of the plane, and \(\omega c\)  = angular velocity of the ball as it passes the center point C of the plane.

"sqrt (5/2)" is the correct option since it matches the expression obtained earlier, i.e., \(\omega f/\omega c = \sqrt{5r/h}\).

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The weight of the wooden pole is 500 N. The correct answer is option A.

To determine the weight of the wooden pole, we can use the principle of moments. The pole is balanced horizontally when the sum of the clockwise moments is equal to the sum of the counter clockwise moments.

When pivoted at a point 2 m from end P, the weight of the pole can be balanced by the moment created by the weight of the pole itself. Let's denote the weight of the pole as W_pole. The clockwise moment created by the weight of the pole is W_pole * 2 Nm.

When pivoted at a point 2 m from end Q, a weight of 500 N is needed to balance the pole horizontally. The counterclockwise moment created by the weight of 500 N is 500 N * 2 Nm.

Since the pole is balanced horizontally in both cases, the clockwise moment and the counter clockwise moment must be equal.

W_pole * 2 Nm = 500 N * 2 Nm

Simplifying the equation:

W_pole = 500 N

Therefore, the weight of the wooden pole is 500 N. The correct answer is option A.

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Jupiter has retained most of its original atmosphere because of its immense size and strong gravitational pull.

Jupiter is the largest planet in our solar system, with a mass of over 300 times that of Earth. Its powerful gravity allows it to hold on to its atmosphere tightly.

Additionally, Jupiter's atmosphere is composed mostly of hydrogen and helium, which are the lightest elements in the universe. This means that they have low escape velocities, and as such, they tend to be held in the planet's gravitational field.

Jupiter's gravity is strong enough to prevent these light gases from escaping into space, thus allowing the planet to retain its atmosphere over time.

Furthermore, Jupiter's strong magnetic field traps charged particles from the solar wind, which also helps to maintain its atmosphere. These particles become ionized in the planet's magnetosphere and can become trapped in the planet's magnetic field.

This creates a radiation belt around Jupiter, which can also affect the planet's atmosphere by causing it to glow and producing auroras.

In summary, Jupiter's large size and strong gravity, as well as its composition and magnetic field, have all contributed to its ability to retain most of its original atmosphere.

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Yes, that statement is correct. The larger container of water will have more heat energy available for heat conduction due to its larger volume and higher mass. This means that it will take longer to heat up or cool down compared to a smaller container with less water. Additionally, the larger surface area of the container also allows for more efficient heat transfer through convection and radiation. Therefore, a larger container of water can be more effective for providing heat to a space or conducting heat through a system.

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The position of the ball attached to the unforced spring with a damping force of 6 times velocity is given by the function \(x(t) = e^{-3}t (sin3t)\). The system is overdamped, and the maximum displacement from equilibrium is 0.1573 meters.

a) Solve for the position of the ball.

The equation of motion of the ball attached to the unforced spring with damping force of 6 times velocity can be written as, \(m(d^{2}x/dt^{2}) + 6(dx/dt) + kx = 0\)

The given values are,

As we are supposed to find the position of the ball, we will solve the differential equation by assuming the position x as the solution and by integrating the given equation two times.

\(m\left(\frac{{d^2x}}{{dt^2}}\right) + 6\left(\frac{{dx}}{{dt}}\right) + kx = 0\)

This is the standard form of a second order homogeneous linear differential equation. The characteristic equation of this differential equation is, \(m^{2} r^{2} + 6mr + k = 0\)

Solving the above quadratic equation, we get, \(r = -3 \pm \sqrt{9 - \frac{4k}{m^2}} / 2m\)

Here, \(k/m = 9/1 = 9\). So, \(r = -3 \pm \sqrt{9 - 36} / 2 = -3 \pm 3i\)

From the above values of r, we can say that the general solution of the differential equation is, \(x(t) = e^{-3t}(C_1\cos(3t) + C_2\sin(3t))\)

Let's find the values of constants C1 and C2 using the initial values of the ball position and velocity.

At

So,

Substituting these values in the general solution of \(x(t),x(t) = e^{-3}t (sin3t)\)

Therefore, the position of the ball as a function of time is given by, \(x(t) = e^{-3}t (sin3t)\).

b) The damping force in the given equation is, b = 6 times the velocity.Since the damping force is greater than the critical damping force \((2\sqrt{m \cdot k})\), the given spring is overdamped.

c) Show that the maximum displacement of the ball from equilibrium is a az meters. To find the maximum displacement of the ball from equilibrium, we can differentiate the position function with respect to time and equate it to zero.

d). \((x(t)) / dt = e^{-3}t (3cos3t - sin3t)\)

When the above derivative of the position function is zero, the position of the ball is at the maximum or minimum from the equilibrium.

Substituting the values of t in the above equation, we get,cos3t = sin3t

Therefore, \(\tan(3t) = 1 \quad t = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{9\pi}{12}, \frac{13\pi}{12}, \frac{17\pi}{12}, \ldots \quad \text{For } t = \frac{\pi}{12}\), the position of the ball is at maximum from equilibrium.

Substituting this value in the position function,\(x(t) = e^{-3t} \sin(3t) \quad x\left(\frac{\pi}{12}\right) = e^{-3\left(\frac{\pi}{12}\right)} \sin\left(\frac{\pi}{4}\right) = 0.1573 \, \text{m}\)

Therefore, the maximum displacement of the ball from equilibrium is \(0.1573\) meters.

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Answer:

78.2 m/time

Explanation:

Wavelength =Speed/Frequency

Speed = 46 x 1.7 =78.2 m/time.

Answer:

3 m/s

Explanation:

v= s/t

v= 9/6

v=3 m/s

Answer:

20 20 20 20 25 points 20 20 20 25 points

Explanation:

To determine the rise in temperature caused by the conversion of mechanical energy into heat in a 90.0 m waterfall, additional information is needed. Specifically, the specific heat capacity of the water and the amount of mechanical energy being converted into heat are required to calculate the temperature rise using the formula Q = mcΔT, where Q is the heat energy, m is the mass of the water, c is the specific heat capacity, and ΔT is the temperature change. Without these values, it is not possible to provide a specific answer.

Once we have all the values, we can substitute them into the formula to find the temperature rise. Without the specific values for mass or other variables, we cannot provide a specific answer.

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The energy that is not transferred into the electrical grid is lost as heat as it moves from system to system.

The electricity that is produced in power stations reach the consumers through a complex system of electric cables and grid. The grid usually means transformers which is used to step up and step down the voltage when necessary.

The first transformer from the power plant steps up the voltage of the current for the purpose of long distance transmissions. Before entering the household another transformer steps down the voltage to avoid short circuiting the electrical appliances. During all these processes the current is carried through electrical lines. Due to its contact with the atmosphere some energy is lost has heat in the air.

Therefore, the energy that is not transferred into the electrical grid is lost as heat as it moves from system to system.

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The thickness of the panel should be 1.82 cm.

                                                        \(\begin{aligned}\\\\\frac{99.2\%}{85.5\%}&=\frac{L}{1.57\,cm}\\\\L&=1.82\,cm\end{aligned}\)

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