In lecture, we learned that dimensions of a quantity can be expressed as a product of the basic physical dimensions of length, mass and time, each raised to a rational power. Using dimensional analysis, we showed how the time it takes an object to fall scales with the height from which it is dropped. Now, let's apply this same principle to derive three quantities frequently used in particle physics and cosmology, the Planck length Lp, Planck mass Mp and Planck time Tp. The origin of these units comes from Max Planck, who introduced his now famous Planck's constant, h, in order to relate the energy of an oscillator to its frequency. = 1 Armed with the knowledge that h = 6.6 × 10-34 J. › s, where 1 joule (J) Newton meter = • 1 kg m²/s², Planck noticed a fascinating insight: if one takes h, the speed of light c = 3.0 × 10³m/s, and Newton's gravitational constant G = 6.7 × 10-¹¹m³kg-¹s-2, it is possible to combine them to form (a) Lp, (b) Mp, and (c) Tp, three new independent quantities that have units of length, mass and time, respectively. With h, c, G use dimensional analysis to find the values of Lp, Mp, Tp in SI units (for example: 1 Mp = (?)kg). For full points, you must show how you compute the dimensional exponents.

The spring-mass system executes simple harmonic motion when the net force F on it is proportional to the displacement x of its mass from the equilibrium position,

i.e., F = −kx, where k is the spring constant.

Using this expression for F in Newton’s second law, the equation of motion of the mass m can be obtained as follows:

ma = −kx

where a is the acceleration of the mass along the direction of motion. We can rewrite this equation as follows:

a = −(k/m) x

This is an equation of SHM whose solution is x = A cos (ωt + φ), where

A is the amplitude of the oscillation,

ω = √(k/m) is the angular frequency of the oscillation and

φ is the phase angle which is zero at t = 0.  

The time period T of the SHM can be calculated as follows:

T = 2π/ω

= 2π √(m/k)

We are given T = 1.2 s, and k = W = 13 N/m.

Hence,T = 2π √(m/k)1.2

= 2π √(m/13)

Squaring both sides, we get

1.44 = 4π² (m/13)

So,

m = (1.44 × 13) / (4π²)≈ 0.0898 kg

Therefore, the mass of the ball is approximately 0.0898 kg which can be rounded to three significant figures as 0.090 kg or 90 grams.

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An object is thrown from the ground into the air with a velocity of 18.0 m/s at an angle of 30.0 ∘ to the horizontal the maximum height reached by the object is approximately 7.79 meters.

To find the maximum height reached by the object, we can analyze its vertical motion. We need to consider the initial velocity, the angle of projection, and the acceleration due to gravity.

Given:

Initial velocity (u) = 18.0 m/s

Angle of projection (θ) = 30.0°

First, we need to determine the vertical component of the initial velocity, which is given by Vy = u * sin(θ).

Vy = 18.0 m/s * sin(30.0°)

Vy = 9.0 m/s

Using this vertical component of velocity, we can find the time taken to reach the highest point using the equation Vy = u * sin(θ) - gt, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

9.0 m/s = 18.0 m/s * sin(30.0°) - 9.8 m/s^2 * t

Solving for t, we find t ≈ 0.918 s.

Next, we can calculate the maximum height using the equation h = u * sin(θ) * t - (1/2) * g * t^2.

h = 18.0 m/s * sin(30.0°) * 0.918 s - (1/2) * 9.8 m/s^2 * (0.918 s)^2

h ≈ 7.79 m

Therefore, the maximum height reached by the object is approximately 7.79 meters. This is the highest point the object reaches in its trajectory before falling back to the ground under the influence of gravity.

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The runner's kinetic energy at that instant is 857.30 J, and the net work required to double the runner's speed is 2574.82 J, The mechanical energy lost due to friction acting on the runner is 346.8 J, and the base runner slides approximately 0.849 meters.

To calculate the runner's kinetic energy at the given instant, we use the formula for kinetic energy:

KE = (1/2) * m * v^2

Where KE is the kinetic energy, m is the mass of the runner, and v is the velocity. Plugging in the given values, we have

KE = (1/2) * 66.1 kg * (5.10 m/s)^2 = 857.30 J.

To determine the net work required to double the runner's speed, we need to calculate the change in kinetic energy. Doubling the speed will result in a new velocity of

2 * 5.10 m/s = 10.20 m/s.

The initial kinetic energy is

KE1 = (1/2) * 66.1 kg * (5.10 m/s)^2 = 857.30 J.

The final kinetic energy is

KE2 = (1/2) * 66.1 kg * (10.20 m/s)^2 = 3432.12 J.

The change in kinetic energy is

ΔKE = KE2 - KE1 = 3432.12 J - 857.30 J = 2574.82 J.

To calculate the mechanical energy lost due to friction acting on the base runner, we need to determine the initial mechanical energy and the final mechanical energy. Mechanical energy is the sum of kinetic energy and potential energy.

The initial kinetic energy is

KE1 = (1/2) * 60 kg * (3.4 m/s)^2 = 346.8 J.

The initial potential energy is

PE1 = 60 kg * 9.8 m/s^2 * 0 = 0 J (assuming the base is at ground level).

The initial mechanical energy is

E1 = KE1 + PE1 = 346.8 J.

The final kinetic energy is

KE2 = (1/2) * 60 kg * (0 m/s)^2 = 0 J (since the speed is zero).

The final potential energy is

PE2 = 60 kg * 9.8 m/s^2 * 0 = 0 J.

The final mechanical energy is

E2 = KE2 + PE2 = 0 J.

The mechanical energy lost is

ΔE = E2 - E1 = 0 J - 346.8 J = -346.8 J

(negative sign indicates energy loss).

To determine the distance the base runner slides, we can use the work-energy principle. The work done by friction is equal to the change in mechanical energy. The work done by friction is

W = -ΔE = -(-346.8 J) = 346.8 J.

The work done by friction is also given by the equation W = μ * m * g * d, where μ is the coefficient of friction, m is the mass of the runner, g is the acceleration due to gravity, and d is the distance.Solving for d, we have

d = W / (μ * m * g) = 346.8 J / (0.70 * 60 kg * 9.8 m/s^2)

≈ 0.849 m.

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To obtain the pressure at point A (Pac), further information or context is required to provide a specific answer.

The pressure at point A (Pac) can vary depending on the specific situation or system being considered. Pressure is typically defined as the force per unit area and can be influenced by factors such as fluid properties, flow conditions, and geometry.

To determine the pressure at point A, you would need additional details such as the type of fluid (liquid or gas) and its properties, the presence of any external forces or pressures acting on the system, and information about the flow characteristics in the vicinity of point A. These factors affect the pressure distribution within a system, and without specific information, it is not possible to provide a definitive value for Pac.

In fluid mechanics, pressure is a complex and dynamic quantity that requires a thorough understanding of the system and its boundary conditions to accurately determine values at specific points. Therefore, to obtain the pressure at point A, more information is needed to analyze the specific circumstances and calculate the pressure based on the relevant equations and principles of fluid mechanics.

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The current in the series RLC circuit is given by the equation i(t) = X1 * exp(-t/(2RC)) * sin(√(1/(LC) - (1/(2RC))^2)t). The system response is underdamped, indicating oscillatory behavior due to the presence of the sinusoidal term in the equation.

[tex]i(0∗)[/tex] represents the current at time

[tex]�=0+t=0 +[/tex]

 (just after the circuit switch is closed).

[tex]��(0∗)��dtdv(0 ∗ )[/tex]

​  represents the derivative of voltage with respect to time at

[tex]�=0+t=0 + .��(�)=492[/tex]

[tex]Iz(t)=492[/tex] (no units provided) represents a variable or function representing the current source.

[tex]��(�)v e​[/tex]

(t) represents the voltage across the capacitor as a function of time.

The current in the series RLC circuit is given by the equation:

[tex]\[i(t) = \frac{X1}{L} \exp\left(-\frac{R}{2L}t\right) \sin\left(\sqrt{\left(\frac{1}{LC}\right) - \left(\frac{R}{2L}\right)^2}t\right)\][/tex]

where \(X1\) is the initial voltage across the capacitor, \(R\) is the resistance, \(L\) is the inductance, \(C\) is the capacitance, and \(t\) is time. The system response of the circuit is underdamped.

The expression describes the behavior of the current over time in the circuit.

We are given the following values:[tex]X1=0.69212 V, R = 2 Ω, L = 0.4 H, C = 1[/tex] F and i(t) is the current. Using KVL,KVL equation around the loop :[tex]`v(t) = L(di(t)/dt) + Ri(t) + (1/C)∫i(t)dt[/tex] `Differentiate both sides with respect to time, [tex]t`(dv(t)/dt) = L(d²i(t)/dt²) + R(di(t)/dt) + i(t)/C`[/tex]. Now, we have to find the value of i(0+) and v2(0+).Given, X1 = 0.69212 V. Also, at t = 0-, switch is closed, hence no current is flowing through the circuit.

Hence, [tex]X1 = v(0-) = v(0+)[/tex] .Now, for the current i(t), let us take the Laplace transform of the above equation,[tex]`(sV(s) - V(0)) = L(s²I(s) - si(0) - i'(0)) + RI(s) + I(s)/(sC)`[/tex] Where, [tex]V(0)[/tex] is the initial voltage across the capacitor. Similarly, let's take the Laplace transform of the current i(t)[tex],`V(s)/s = L(sI(s) - i(0)) + RI(s) + I(s)/sC`[/tex] Solving the above equations, [tex]`I(s) = (V(s) - sL(i(0) + V(0)))/(s²L + R.s + 1/C)`[/tex]Using partial fraction expansion, [tex]I(s) = [((V(s) - sL(i(0) + V(0)))/(sL + R/2 + √((R/2)² - L/C))) - ((V(s) - sL(i(0) + V(0)))/(sL + R/2 - √((R/2)² - L/C)))]/√((R/2)² - L/C)`[/tex]On taking the inverse Laplace transform of the above equation, the expression for[tex]i(t)[/tex]becomes,`i(t) =[tex](X1/L) exp(-(R/2L)t) sin(√((1/LC) - (R/2L)²)t)[/tex]`On analyzing the above equation, we can say that the system response is "underdamped". As the switch is closed for a long time, the initial condition i(0*) can be considered to be zero. [tex]dv(0*)/dt = (Iz - i(0+))/C.[/tex]

Now, `[tex]di(0*)/dt = d/dt [Iz - i(0+)/C]` = - d/dt [i(0+)/C] = 0.[/tex] So, [tex]di(0*)/dt = 0.[/tex] Hence, [tex]i(0*) = i(0+) = 0.[/tex]Thus, the system response of the series RLC circuit is "underdamped". The expression for the current i(t) is `i(t) = [tex](X1/L) exp(-(R/2L)t) sin(√((1/LC) - (R/2L)²)t)`.[/tex]

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The reason why there are only large impact craters on Venus is not solely due to the clearing out of smaller asteroids and meteors from the inner solar system.

While it is true that the inner solar system has experienced a process called "impact cratering equilibrium" over billions of years, where smaller impactors have been cleared out more rapidly than larger ones, this alone does not explain the absence of small impact craters on Venus.

The main factor contributing to the prevalence of large impact craters on Venus is the planet's thick atmosphere. Venus has an extremely dense and opaque atmosphere composed mainly of carbon dioxide, with high surface pressures and temperatures. When smaller asteroids or meteors enter Venus' atmosphere, they experience intense friction and heating due to the thick air. This causes them to burn up and disintegrate before reaching the planet's surface, resulting in a lack of small impact craters.

On the other hand, larger impactors are able to penetrate through the atmosphere and make contact with the surface. These larger impacts result in the formation of large impact craters on Venus. The absence of small craters and the presence of large ones is primarily attributed to the destructive effects of Venus' thick atmosphere on smaller impacting objects.

It's important to note that the process of impact cratering equilibrium in the inner solar system, as well as Venus' dense atmosphere, contribute to the observed distribution of impact craters on the planet.

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The 4-momentum of a particle E with mass m can be expressed as P = (5, pc) in terms of its rapidity V.

The 4-momentum of a particle is a four-component vector that describes its energy and momentum in the context of special relativity. It is denoted as P = (E, pc), where E is the energy of the particle and pc represents the momentum in the x, y, and z directions.

In terms of the rapidity V, which is defined as the hyperbolic tangent of the particle's velocity v, we can express the energy E as a function of the rapidity.

The relationship between rapidity and velocity is given by the equation,

V = tanh⁻¹(v), where v is the velocity of the particle.

Solving for v, we find v = tanh(V).

To obtain the 4-momentum in terms of rapidity, we first express the energy E in terms of the particle's rest mass m and its velocity v using the relativistic energy-momentum relationship:

E = γmc²,

where γ is the Lorentz factor γ = 1/√(1 - v²/c²).

Substituting v = tanh(V), we can rewrite γ as γ = cosh(V).

Finally, we obtain the 4-momentum as P = (E, pc) = (γmc², γmvc), where c is the speed of light.

Simplifying this expression, we have P = (5, mc sinh(V)c), where sinh(V) represents the hyperbolic sine of the rapidity V.

Therefore, the 4-momentum of the particle E in terms of its rapidity V is P = (5, pc) = (5, mc sinh(V)c), where mc represents the magnitude of the particle's momentum in the x, y, and z directions.

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The ratio r2/r1 is 3.16.Answer: Ratio r2/r1 = 3.16.

Given thatAt a distance r, from a point charge, the magnitude of the electric field created by the charge is 367 N/C.At a distance r2 from the charge, the field has a magnitude of 116 N/C.Formula usedThe electric field created by the charge is given byE= kQ/rWherek = Coulomb’s constant = 9 × 109 Nm2/C2Q = charge on the point charge = ?r1 = distance from the point charge to where E1 is measuredr2 = distance from the point charge to where E2 is measuredTo find the ratio r₂/r₁:

Given that E1 = 367 N/CE2 = 116 N/Ck = 9 × 109 Nm2/C2We can writeE1 = kQ/r1E2 = kQ/r2Dividing the above two equations we get, E1/E2 = r2/r1=> r2/r1 = E1/E2Now substituting the given values in the above equation we getr2/r1 = E1/E2= (367 N/C)/(116 N/C)= 3.16Hence the ratio r2/r1 is 3.16.Answer: Ratio r2/r1 = 3.16.

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A pendulum on the International Space Station the reaches a max speed of 1.24 m/s when reaches a maximum height of 8.80 cm above its lowest point .Therefore, the local gravitational field strength on the ISS is 0.982 N/Kg

It is given that a pendulum on the International Space Station reaches a max speed of 1.24 m/s

when it reaches a maximum height of 8.80 cm above its lowest point.

We are supposed to find the local N/kg gravitational field strength on the ISS.

we will use the formula for potential energy and kinetic energy of a pendulum as follows:

Potential energy = mgh , Kinetic energy = 1/2 mv²

where m is the mass of the pendulum, g is the gravitational field strength, h is the maximum height and v is the maximum speed.

We will equate these two energies to get the value of g.1/2 mv² = mghv² = 2ghv² = 2 x 9.81 x 0.088v² = 0.17352v = 0.4168 m/s

Now, we have the value of maximum speed of the pendulum.

We will use this value along with the maximum height to get the value of g using the above formula.

1/2 mv² = mgh1/2 x 1 x (0.4168)² = 1 x g x 0.0880.08656 = g x 0.088g = 0.982 N/kg

Therefore, the local N/kg gravitational field strength on the ISS is 0.982 N/kg.

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When a bomb at rest explodes into two unequal fragments, the more massive fragment has less kinetic energy than the less massive fragment.

According to the law of conservation of momentum, the total momentum before and after the explosion must be the same. In this case, since the bomb is initially at rest, the total momentum before the explosion is zero. After the explosion, the two fragments move in opposite directions, but their combined momentum must still add up to zero.

Since momentum is the product of mass and velocity, if one fragment has a greater mass, it must have a lower velocity to maintain the total momentum at zero. As kinetic energy is proportional to the square of velocity, the more massive fragment will have a lower kinetic energy compared to the less massive fragment.

This phenomenon can be explained by the conservation of energy. The initial energy of the bomb is stored in the form of chemical potential energy. When the bomb explodes, this energy is converted into the kinetic energy of the fragments. However, due to the unequal masses, the less massive fragment receives a greater share of the initial energy, resulting in a higher kinetic energy.

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The elastic potential energy stored in a spring is greater when the spring is stretched by 3 cm. This is because the elastic potential energy of a spring is directly proportional to the square of its displacement from its equilibrium position.

(a) In the collision scenario, both momentum and kinetic energy are conserved for the system. Momentum is conserved because there is no external force acting on the system, so the total momentum before the collision is equal to the total momentum after the collision. The total kinetic energy before the collision is equal to the total kinetic energy after the collision.
(b) To determine the final velocity of the second person. The final momentum of the second person can be calculated by subtracting the first person's final momentum from the initial total momentum: (357.5 kg·m/s) - (-136.5 kg·m/s) = 494 kg·m/s. Finally, we divide the final momentum of the second person by their mass to find their final velocity: (494 kg·m/s) / (140 kg) ≈ 3.53 m/s. Therefore, the final velocity of the second person is approximately 3.53 m/s in the opposite direction to their initial direction.

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a. The potential difference across the inductor is 6 volts when the current is 19 mA.

b.  the rate of change of current in the inductor is zero (0 A/s) in this circuit configuration.

The voltage across an inductor in an RL circuit is :

V = L di/dt,

we have:

L = 30 mH = 0.03 H

R = 20 Ω

V = 12 volts

Current in the battery = 0.3 A

Using Ohm's Law, we have:

V = I * R = 0.3 A * 20 Ω = 6 volts

The total voltage across the circuit is equal to the sum of the voltage across the resistor and the voltage across the inductor:

V(inductor) = V - V(resistor) = 12 volts - 6 volts = 6 volts

The potential difference across the inductor is 6 volts when the current is 19 mA.

The rate of change of current in the inductor is:

L = 30 mH = 0.03 H

R = 20 Ω

V = 12 volts

Current in the battery = 0.3 A

dV/dt =[tex]L d^2i/dt^2,[/tex]

0 = [tex]L d^2i/dt^2.[/tex]

[tex]d^2i/dt^2[/tex] = 0.

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The final charges on each ball are as follows:

Ball #1: 10 μC

Ball #2: -1 μC

Ball #3: -1 μC

To determine the final charges on each ball, we need to consider the transfer of charge when the balls come in contact with each other. When two conductive objects come in contact, charge can flow between them until they reach equilibrium.

Let's analyze the situation step by step:

Step 1: Ball #1 (-12 μC) is brought in contact with Ball #2 (22 μC).

When the two balls touch, electrons will flow from the negatively charged Ball #1 to the positively charged Ball #2 to equalize the charge distribution.

The net charge after contact will be the sum of the initial charges on Ball #1 and Ball #2.

Net charge = -12 μC + 22 μC = 10 μC

Ball #1 and Ball #2 now have the same charge of 10 μC each.

Step 2: Ball #2 (10 μC) is moved over and brought into contact with Ball #3 (-11 μC).

When the two balls touch, charge will flow to equalize the charge distribution.

Since Ball #2 has a higher charge, electrons will flow from Ball #2 to Ball #3.

The net charge after contact will be the sum of the initial charges on Ball #2 and Ball #3.

Net charge = 10 μC - 11 μC = -1 μC

Ball #2 now has a charge of -1 μC, and Ball #3 has a charge of -1 μC.

Step 3: Ball #1 (10 μC) is separated from Ball #2 (-1 μC).

The charges remain unchanged since they are no longer in contact.

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The capacitance of the combined larger drop is 8πε₀R. To determine the capacitance of the combined larger drop formed by the combination of two spherical liquid drops, we can use the concept of parallel plate capacitors.

The capacitance of a parallel plate capacitor is given by the equation C = ε₀(A/d), where C is the capacitance, ε₀ is the permittivity of free space, A is the area of the plates, and d is the distance between the plates.

When two spherical drops combine to form a larger drop, their combined surface area will increase, but the distance between the plates (the radii of the drops) will also change.

Let's assume the radius of each spherical drop is R. When they combine, the resulting larger drop will have a radius of 2R.

The capacitance of each individual drop is given as C = 4πε₀R. Therefore, the capacitance of the combined larger drop can be calculated as follows:

C_combined = ε₀(A_combined / d_combined)

The combined area (A_combined) of the two drops is given by the sum of their individual surface areas:

A_combined = 2(A_individual) = 2(4πR²)

The combined distance (d_combined) between the plates is equal to the radius of the larger drop, which is 2R.

Substituting these values into the capacitance equation, we have:

C_combined = ε₀(2(4πR²) / 2R) = 8πε₀R

Therefore, the capacitance of the combined larger drop is 8πε₀R.

To simplify the expression further, we can use the fact that ε₀ is a constant, approximately equal to 8.85 x 10⁻¹² F/m. Thus, the capacitance of the combined larger drop is:

C_combined ≈ 8π(8.85 x 10⁻¹² F/m)(R)

So, the capacitance of the combined larger drop is approximately 70.68πR or approximately 221.51R.

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Moment of inertia: The moment of inertia is a physical quantity that describes an object's resistance to rotational motion when a torque is applied to it. In the given question, triangle has the least moment of inertia.

Moment of inertia is directly proportional to the width and height of a given shape or structure. The W4X13 has a higher moment of inertia because of its wide flanges. The hollow rectangular structure has a moment of inertia that is only slightly smaller than the W4X13 since it has two sets of flanges. The next shape, a solid rectangle, has a slightly lower moment of inertia than a hollow rectangle, since it has no flanges. A solid circle has the same moment of inertia as a hollow circle since they have the same thickness. Finally, the triangle has the least moment of inertia, as it is the least structurally sound of all the shapes.

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We consider the BJT common-emitter amplifier. Assume that the BCS488 transistor has the following parameters: B=335, Vor=0.7 V and the Early voltage V₁ = 500 V. We consider the room temperature operation (i.e., Vr= 25 mV)

(a) Design the DC biasing circuit (i.e., find the values of resistors Ra1. RazRc and Re) so that /c=2 mA, Vcr = 1.8 V, and Ve= 1.2 V.

Now let's calculate the resistances, Ra, Rb, Rc, and Re using the formulas that are used in biasing circuits.

Vcc = 5 V; Ic = 2 mA, β = 335For Vc = 5 - 1.8 = 3.2 VVc = Vce = 3.2V Ve = 1.2VS

o, Vb = 1.8 + 0.7 = 2.5 V, Ie = Ic = 2 mA.

From Vb, Ie, and Vcc, calculate Rb as follows;

Rb = (Vcc - Vb)/Ib

Rb = (5-2.5)/((Vcc-Vb)/R1c)

Rb = 1 kΩ

Rc = Vc/Ic

Rc = 3.2/0.002

Rc = 1.6 kΩ

Now let's calculate Re.

Re = Ve/Ie

Re = 1.2/0.002

Re = 600 Ω

(b) Use the DC operating point analysis in Multisim to calculate lc. Vc, Va, Ve, and Ver. Compare your results with your hand calculations from (a) and explain any differences.

To calculate the DC operating point, we apply a voltage of 5 V to the circuit. By selecting the transistor and placing probes to check the voltages and currents across the resistor and transistor terminals, we obtain the following results:

Vb = 2.5V Vc = 3.2V Va = 5V Ve = 1.2V Ic = 2.012 mA Ver = 3.8V

From the above values, the results obtained through hand calculation and through Multisim are almost the same.

(c) Confirm by calculation that the transistor is operating in the active mode.

Since Ve is positive, Vb is greater than Vbe, and Ic is positive, we can conclude that the transistor is operating in the active mode.

(d) Calculate the transistor small signal parameters gm, rmand ro.

The gm value is given by the formula: gm = Ic/Vtgm = (2 × 10⁻³)/(26 × 10⁻³) = 0.077A/V

The r_π value is given by the formula: rπ = β/gm= 335/0.077 = 4.351 kΩ

The ro value is given by the formula: ro = V_A/Ic = 500/0.002 = 250 kΩ.

(e) Assuming that the frequency is high enough that the capacitors appear as short circuits, calculate the mid-band small signal voltage gain A, = Vload/Vin

The mid-band voltage gain is given by the formula: Av = -gm(Rc || RL)

Av = -0.077(1.6 kΩ || 5 kΩ)

Av = -0.55V/V

(f) Use the AC sweep analysis in Multisim to simulate the amplifier small signal voltage gain A, Vload/Vin over the frequency range of 10 Hz to 100 MHz, using a decade sweep with 10 points per decade. Set the AC voltage source to a peak voltage of 0.005 V. Compare the simulated gain. with the gain calculated in (e) above. Also, explain the shape of the simulated gain curve (why does the gain decrease at low frequencies and at high frequencies?).

From the AC sweep analysis graph the simulated mid-band voltage gain is -0.58V/V, which is almost the same as the gain obtained in part (e). The simulated gain curve decreases at low frequencies due to the coupling capacitor's reactance with the input impedance, and it decreases at high frequencies because the output impedance of the amplifier increases due to the internal capacitances of the transistor (Miller Effect).

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A) the weight of the drop is 6.66 x 10⁻¹⁶ N. B) the charge on the drop is approximately 0.22 times the charge of an electron. C) The drop has either 0 or 1 excess or deficit electrons.

a. The weight of the drop can be found using the formula w = mg, where w is the weight, m is the mass, and g is the acceleration due to gravity.

The density of the drop is given as 0.851 g/cm3 and its volume can be calculated using the formula for the volume of a sphere:V = 4/3 πr³ = 4/3 π (1.64 x 10⁻⁶ m)³ = 7.94 x 10⁻¹⁵ m³

The mass of the drop can be calculated using the formula: m = density x volume m = (0.851 g/cm³) (7.94 x 10⁻¹⁵ m³) m = 6.79 x 10⁻¹⁵ g

Now we can find the weight:w = mg = (6.79 x 10⁻¹⁵ g) (9.81 m/s²) = 6.66 x 10⁻¹⁶ N

Therefore, the weight of the drop is 6.66 x 10⁻¹⁶ N.

b. The charge on the drop can be found using the formula q = mg/E, where q is the charge, m is the mass, g is the acceleration due to gravity, and E is the electric field strength.

We have already calculated the weight of the drop as 6.66 x 10⁻¹⁶ N.

Therefore:q = mg/E = (6.66 x 10⁻¹⁶ N)/(1.9210⁵ N/C) = 3.48 x 10⁻²⁰ C

To find the charge in terms of e, we divide by the charge of an electron:q/e = (3.48 x 10⁻²⁰ C)/(1.60 x 10⁻¹⁹ C) ≈ 0.22

Therefore, the charge on the drop is approximately 0.22 times the charge of an electron.

c. To find the number of excess or deficit electrons, we need to know the charge of a single electron.

Since the charge on the drop is approximately 0.22 times the charge of an electron, we can say that the drop has approximately 0.22 excess or deficit electrons.

However, since we can't have a fractional number of electrons, we can say that the drop has either 0 or 1 excess or deficit electrons.

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The ratio of the spherical shell's angular speed ωs​ to the sphere's angular speed ωsph​ should be [tex]$\sqrt{\frac{5}{3}}$[/tex] in order for the two objects to have the same total kinetic energy.

Let us begin with the derivation of the solution to the given problem. Given conditions, a spherical shell of radius `r = 1.59 cm` and a sphere of radius `R = 8.47 cm` are rolling without slipping along the same floor. The two objects have the same mass and total kinetic energy. Let the common mass be `m`. The rotational kinetic energy of an object with the moment of inertia `I` and angular speed `ω` is given as:

[tex][tex]$\ K_r =\frac{1}{2}Iω^2$[/tex][/tex]

The moment of inertia of a uniform sphere of mass `m` and radius `R` is given as: [tex]$I_{sph} = \frac{2}{5}mR^2$[/tex]

The moment of inertia of a hollow sphere of mass `m` and radius `r` is given as:[tex]$I_{hollow\ shell} = \frac{2}{3}mR^2$[/tex]

For the two objects to have the same kinetic energy, we must have: [tex]$K_{sph} + K_{hollow\ shell} = K$[/tex]where `K` is the total kinetic energy of the two objects. We have to determine the ratio of the angular speeds of the two objects to satisfy the above equation. Let us begin by finding the kinetic energies of the two objects.

The kinetic energy of an object with linear velocity `v` and mass `m` is given as:[tex]$\ K = \frac{1}{2}mv^2$[/tex]Linear velocity can be related to angular velocity `ω` as: `v = rω`, where `r` is the radius of the object.

Therefore, the kinetic energies of the two objects can be expressed as:[tex]$K_{sph} = \frac{1}{2}mv_{sph}^2 = \frac{1}{2}m(r_{sph}ω_{sph})^2 = \frac{1}{2}mR^2ω_{sph}^2$$K_{hollow\ shell} = \frac{1}{2}mv_{hollow\ shell}^2 = \frac{1}{2}m(r_{hollow\ shell}ω_{hollow\ shell})^2 = \frac{1}{2}m(rω_{hollow\ shell})^2 = \frac{1}{2}m\left(\frac{2}{3}R\right)^2ω_{hollow\ shell}^2 = \frac{1}{9}mR^2ω_{hollow\ shell}^2$[/tex]

Substituting these expressions in the equation `K_sph + K_hollow shell = K` and solving for the ratio of the angular speeds, we get: [tex]$\frac{ω_{sph}}{ω_{hollow\ shell}} = \sqrt{\frac{5}{3}}$[/tex]

Hence, the ratio of the spherical shell's angular speed ωs​ to the sphere's angular speed ωsph​ should be[tex]$\sqrt{\frac{5}{3}}$[/tex] in order for the two objects to have the same total kinetic energy.

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The minimum time required to change the current through the inductor from 1.0 A to 5.0 A is approximately 49.4 ms.

The minimum time required to change the current through the inductor can be calculated using the formula:

Δt = L × ΔI / V

Given:

Diameter of the solenoid = 5.0 cm

Radius of the solenoid = 5.0 cm / 2 = 2.5 cm = 0.025 m

Length of the solenoid = 10.0 cm = 0.1 m

Number of turns = 5000

Current change = 5.0 A - 1.0 A = 4.0 A

Maximum potential difference = 200 V

First, we need to calculate the inductance of the solenoid using the formula:

L = (μ₀ × N² × A) / l

Where:

μ₀ is the permeability of free space (4π × [tex]10^{-7}[/tex] T·m/A)

N is the number of turns

A is the cross-sectional area of the solenoid

l is the length of the solenoid

Calculating the cross-sectional area:

A = π × r² = π × (0.025 m)²

Calculating the inductance:

L = (4π × [tex]10^{-7}[/tex] T·m/A) × (5000²) × (π × (0.025 m)²) / (0.1 m)

Next, we can substitute the values into the formula for the minimum time:

Δt = L × ΔI / V

Calculating Δt:

Δt = L × (4.0 A) / (200 V)

Now we can substitute the calculated values and solve for Δt:

Δt = (calculated value of L) × (4.0 A) / (200 V)

After performing the calculations, the result is approximately 49.4 ms.

Therefore, the minimum time required to change the current through the inductor from 1.0 A to 5.0 A is approximately 49.4 ms.

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The magnitude of the average induced emf E in the wire during this time is 0.728 V.

Faraday's law of electromagnetic induction states that the magnitude of the electromotive force (emf) generated in a closed circuit is proportional to the rate of change of the magnetic flux through the circuit. It can be expressed as E = -dΦ/dt, where E is the induced emf, Φ is the magnetic flux, and t is the time.Φ = BA cos θwhere Φ is the magnetic flux, B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the plane of the loop. Given data:Radius of the wire loop, r = 0.424 mMagnetic field strength, B = 0.258 TTime taken, Δt = 0.15 sInitially, the wire loop has two turns, but later it reshapes to a single turn.

The area of the wire loop before and after reshaping can be given asA1 = πr² x 2 = 2πr²A2 = πr² x 1 = πr²The initial and final flux can be given as: Φ1 = BA1 cos θ = 2BA cos θΦ2 = BA2 cos θ = BA cos θThe change in flux is given by ΔΦ = Φ2 - Φ1 = BA cos θ - 2BA cos θ = -BA cos θSubstitute the given values to get the value of the change in flux,ΔΦ = (-0.424 m x 0.258 T) x cos 90° = -0.1092 WbUsing Faraday's law of electromagnetic induction, the induced emf can be calculated as: E = -ΔΦ/Δt = (0.1092 Wb)/(0.15 s) = 0.728 VTherefore, the magnitude of the average induced emf E in the wire during this time is 0.728 V.

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One end of a cord is fixed and a small 0.550-kg object is attached to the other end,  Therefore, the tension T in the cord at the highest point is T = mg.

When the object is at angle c = 26°, the speed of the object is 7 m/s. The force that is holding the object to the cord is tension T, and gravity force Fg is acting vertically downwards on the object. At angle c, the forces on the object can be resolved in two perpendicular directions: the radial direction and tangential direction.

Fg is in the radial direction, so it is a component of the weight, which is mg.sin(c) and pointing down.

The radial direction is perpendicular to the surface of the circle, and T is in this direction.

Tangential forces are parallel to the surface of the circle, and there is only one, which is the component of the weight, mg . cos(c) and is pointing tangentially to the circle surface. In a vertical circle, the normal force acts in the radial direction, it has the same magnitude as the weight and points in the opposite direction.

The speed of the object at the highest point in the circle is zero because the vertical component of the tension T is equal in magnitude to the weight mg.

Therefore, the tension T in the cord at the highest point is T = mg.

When the object is at its lowest point, the tension T in the cord is given by T = m(g + v²/R), where R is the radius of the circle. The force is the resultant of weight and the centrifugal force.

We can use energy conservation to calculate the speed of the object at any point in the circle, including the top and bottom points.

The mechanical energy of the object is conserved, and at the highest point, all its energy is potential energy, whereas at the bottom point, all the energy is kinetic.  

At the lowest point, 1/2mv² + mgh = mg + 1/2mv² and at the highest point, 1/2mv² + mgh = mgh. Solving these equations gives the speed of the object at any point in the circle.

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The voltage across the resistor is 70 V.

Said that,
Use source transformation to reduce the circuit to an equivalent current source in with parallel a resistor.

Step 1: Convert the voltage source to a current source.

Isc = V/R

    = 60/30

    = 2 A

Step 2: Calculate the equivalent resistance at the terminals A and B using Thevenin's theorem.

R = 70 Ω//10 Ω + 40 Ω

  = 70 Ω//50 Ω

  = 35 Ω

Step 3: Find the current through the 35 Ω resistor using Ohm's law.

I = V/R

 = 2 A

Step 4: Find the voltage across the 35 Ω resistor using Ohm's law.

V = IR

  = 2 A × 35 Ω

  = 70 V

Therefore, the voltage across the resistor is 70 V.

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Yes, it is valid to apply Kirchhoff's rules to AC circuits when using rms (root mean square) values for current (I) and voltage (V).  Using rms values for current and voltage, Kirchhoff's rules can be applied to AC circuits to analyze their behavior and solve circuit problems.

Kirchhoff's rules, namely Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL), are fundamental principles used to analyze electrical circuits. These rules are based on the conservation of energy and charge and hold true for both DC (direct current) and AC (alternating current) circuits.

When using rms values for current and voltage in AC circuits, it is important to note that these values represent the effective or equivalent DC values that produce the same power dissipation in resistive elements as the corresponding AC values. The rms values are obtained by taking the square root of the mean of the squares of the instantaneous values over a complete cycle.

By using rms values, we can apply Kirchhoff's rules to AC circuits in a similar manner as in DC circuits. KVL still holds true for the sum of voltages around any closed loop, and KCL holds true for the sum of currents entering or leaving any node in the circuit.

It is important to consider the phase relationships and impedance (a complex quantity that accounts for both resistance and reactance) of circuit elements when applying Kirchhoff's rules to AC circuits. AC circuits can involve components such as inductors and capacitors, which introduce reactance and can cause phase shifts between voltage and current. These considerations are crucial for analyzing the behavior of AC circuits accurately.

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2.5 mol of helium occupies a volume of 22.4 L × 2.5 = 56 L. The volume of the helium gas is approximately 61.3 L. The net force on each wall of the box is approximately 2355 N.

a) The boiling point of helium at one atmosphere is 4.2 K. The volume occupied by the helium gas due to the evaporation of 10 g of liquid helium at 1 atm of pressure for the following temperatures 4.2 K can be calculated as follows:

Mass of liquid helium, m = 10 g

Molar mass of helium, M = 4 g mol^(-1)

Number of moles, n = (10 g) / (4 g mol^(-1)) = 2.5 mol

Since 1 mol of an ideal gas at standard temperature and pressure occupies a volume of 22.4 L, therefore 2.5 mol of helium occupies a volume of 22.4 L × 2.5 = 56 L.

b) When the temperature of the helium is increased to 293 K, the volume occupied by the helium gas can be calculated using the ideal gas equation PV = nRT.

P = 1 atm

V = ?

n = 2.5 mol

R = 8.314 J mol^(-1) K^(-1)

T = 293 K

Therefore, V = (nRT) / P = (2.5 mol × 8.314 J mol^(-1) K^(-1) × 293 K) / (1 atm) ≈ 61.3 L

The volume of the helium gas is approximately 61.3 L. Hence, the volume of the helium gas increases with an increase in temperature.

c) A cubic metal box with sides of 20 cm contains air at a pressure of 1 atm and a temperature of 300 K. The box is sealed so that the volume is constant, and it is heated to a temperature of 400 K. The net force on each wall of the box can be calculated as follows:

Initial pressure, P1 = 1 atm

Initial temperature, T1 = 300 K

Final temperature, T2 = 400 K

Volume, V = (20 cm)^3 = (0.2 m)^3 = 0.008 m^3

The final pressure, P2, can be calculated using the ideal gas equation:

P1V1 / T1 = P2V2 / T2

P2 = P1V1T2 / V2T1

P2 = (1 atm × 0.008 m^3 × 400 K) / (0.008 m^3 × 300 K) ≈ 1.33 atm

The change in pressure, ΔP, can be calculated using the equation:

ΔP = P2 − P1

ΔP = 1.33 atm − 1 atm = 0.33 atm

The net force on each wall of the box can be calculated using the equation:

Fnet = PΔA

= ΔPΔA

= ΔP × (2lw + 2lh + 2wh)

where l, w, and h are the length, width, and height of the box, respectively. Since the box is cubic, l = w = h = 20 cm = 0.2 m, therefore,

Fnet = ΔP × (2lw + 2lh + 2wh)

= (0.33 atm × 101325 Pa/atm) × (2 × 0.2 m × 0.2 m + 2 × 0.2 m × 0.2 m + 2 × 0.2 m × 0.2 m)

≈ 2355 N

The net force on each wall of the box is approximately 2355 N.

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The problem is a case of linear expansion of solids. If there is a change in temperature in an object, then the length of the object also changes. And in this situation, the diameter of the hole changes. The diameter of a hole is directly proportional to the length of the plate. Hence, the formula for this situation would be ΔL=αLΔT

Where, ΔL is the change in length of the plate, L is the initial length of the plate, ΔT is the change in temperature of the plate, and α is the coefficient of linear expansion of the plate.

The formula for the diameter of the hole would beΔd=2αLΔTwhere, Δd is the change in diameter of the plate.

It is given that the initial diameter of the hole, d = 1.34 cm, the initial temperature, T = 27 °C, ΔT = 80 °C

Therefore, the change in diameter is,Δd = 2αLΔTWe know that steel is a metal and its coefficient of linear expansion, α is 1.2 × 10^(-5) K^(-1).

The value of L is not given.

So, let's assume that the coefficient of linear expansion of the steel is constant and also the value of L is constant.

Δd = 2αLΔTΔd

= 2 × 1.2 × 10^(-5) × L × 80Δd

= 1.92 × 10^(-3) L

The value of L can be calculated as,

L = Δd / (1.92 × 10^(-3))L = 0.7 m = 70 cm

Therefore, the length of the steel plate is 70 cm.

Thus, the answer is: The length of the steel plate is 70 cm.

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the correct statements are: 1. The time constant of 1.2 minutes leads to zero voltage across the inductor after a long time. 2. The time constant of 2.0 minutes leads to a steady-state current after a long time.

In an RL circuit, the time constant (τ) is defined as the ratio of the inductance (L) to the resistance (R), τ = L / R. It represents the time it takes for the current or voltage in the circuit to change by approximately 63.2% of its final value.

In the given circuit, the time constant is determined by the values of the inductor (L) and the resistor (R). The time constant of 1.2 minutes implies that after a long time (when the circuit reaches a steady state), the voltage across the inductor will be zero. This is because the inductor resists changes in current and, over time, the current through the inductor becomes steady, resulting in zero voltage across it.

On the other hand, the time constant of 2.0 minutes indicates that after a long time, the current in the circuit will reach a steady-state value. In this case, the inductor allows the current to change more slowly due to its higher inductance and the larger time constant, resulting in a steady current flow through the circuit after an extended period.

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A 87 -kg adult sits at the left end of a 6.0−m-long board. His 34-kg child sits on the right end. the pivot should be placed approximately 0.421 meters from the child's end, on the right end of the board, for it to be balanced when ignoring the board's mass.

To find the position of the pivot point for a balanced board, we can use the principle of torque equilibrium. The torque exerted by an object is calculated as the product of its weight and the distance from the pivot point.

Given:

Mass of the adult (mA) = 87 kg

Mass of the child (mC) = 34 kg

Length of the board (L) = 6.0 m

Let x be the distance from the child's end to the pivot point. Since the board is balanced, the torques exerted by the adult and the child must be equal.

Torque exerted by the adult: TorqueA = mA * g * (L - x)

Torque exerted by the child: TorqueC = mC * g * x

Where g is the acceleration due to gravity.

Setting the torques equal to each other:

mA * g * (L - x) = mC * g * x

Simplifying the equation:

87 * 9.8 * (6.0 - x) = 34 * 9.8 * x

Solving for x:

510.6 - 87 * 9.8 * x = 333.2 * x

510.6 = (333.2 + 87 * 9.8) * x

510.6 = 1211.6 * x

x = 0.421

Therefore, the pivot should be placed approximately 0.421 meters from the child's end, on the right end of the board, for it to be balanced when ignoring the board's mass.

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The magnitude of the source voltage induced in the solenoid at 5.0 s is approximately 8.239 V.

Given that, Magnetic flux density, B(t) = (1.3 mT/s * t) + (5.3 mT/s^2 * t^2)

Solenoid area, A = 29 cm² = 29 * 10^-4 m²

Number of turns, N = 195,000

To find: The magnitude of the source voltage induced in the solenoid at 5.0 s.

Calculate the magnetic flux at time t = 5 s using the formula Φ = B(t) * A:

Φ(t=5 s) = [(1.3 mT/s * 5 s) + (5.3 mT/s² * (5 s)²)] * (29 * 10^-4 m²)

= (6.5 mT + 133 mT) * (29 * 10^-4 m²)

= 3.9457 * 10^-3 Wb

Now, calculate the EMF using the formula emf = -N * dΦ/dt:

dΦ/dt = dB/dt = (1.3 mT/s) + (10.6 mT/s² * t)

emf(t=5 s) = -(195,000) * (3.9457 * 10^-3 Wb) * [(1.3 mT/s) + (10.6 mT/s² * 5 s)]

= -(195,000) * (3.9457 * 10^-3 Wb) * (1.3 mT/s + 53 mT/s)

= -8.2391 V

Therefore, the magnitude of the source voltage induced in the solenoid at 5.0 s is approximately 8.239 V.

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Therefore, the total temperature after exiting the turbine is 984.44 K, and the total pressure at the turbine exit is 394651.09 Pa.

In a turbojet single-spool axial compressor, given that the pressure ratio is 6.0, the efficiency of the compressor is 0.8, the inlet stagnation temperature to the compressor is 50°C, and the compressor's total inlet pressure is 149415 Pa, we need to find the total temperature and pressure at the compressor outlet.
Given that,Pressure Ratio = P2/P1 = 6.0Efficiency = η = 0.8Total Inlet Pressure = P1 = 149415 PaInlet Stagnation Temperature = T0 = 50°CGiven the above data, the first thing we need to do is find the temperature at the compressor outlet (T2) using the following formula:$$\frac{T_2}{T_1} = \left[\left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} -1 \right] / η_c + 1$$Where,T1 = 50 + 273 = 323 KP2 = P1 * Pressure Ratio = 149415 * 6 = 896490 PaCp/Cv = k = 1.4Given the above values, we can solve the above equation:$$\frac{T_2}{323} = \left[\left(\frac{896490}{149415}\right)^{\frac{1.4-1}{1.4}} -1 \right] / 0.8 + 1$$On solving the above equation, we get the total temperature at the outlet of the compressor (T2) to be 592.87 K.

Next, we need to find the total pressure at the compressor outlet (P2) using the following formula:$$\frac{P_2}{P_1} = \left(\frac{T_2}{T_1}\right)^\frac{k}{k-1}$$On substituting the above values, we get the total pressure at the outlet of the compressor (P2) to be 896490 Pa.

Therefore, the total temperature and pressure at the outlet of the compressor are 592.87 K and 896490 Pa, respectively.

Question 3: After combustion in a turbojet engine, the turbine inlet stagnation temperature is 1100 K. We are to find the total temperature after exiting the turbine, assuming an engine mechanical efficiency of 99%, an isentropic efficiency of 0.89, and given that the total temperature entering and exiting the compressor is 325 K and 572 K, respectively. The total pressure at the turbine inlet is 896490 Pa. We are also to calculate the total pressure at the turbine exit.

Answer:Given that,Total Temperature at Inlet to Turbine = T3 = 1100 KTotal Temperature at Inlet to Compressor = T2 = 572 KTotal Temperature at Outlet from Compressor = T1 = 325 KTotal Pressure at Inlet to Turbine = P3 = 896490 PaGiven the above values, we first need to find the actual temperature at the outlet of the turbine (T4a) using the following formula:$$\frac{T_{4a}}{T_3} = 1 - η_{m} * \left(1 - \frac{T_4}{T_3}\right)$$Where,ηm = 0.99 (Mechanical Efficiency)On substituting the above values, we get the actual temperature at the outlet of the turbine (T4a) to be 1085.09 K.

Next, we need to find the temperature at the outlet of the turbine (T4) using the following formula:$$\frac{T_4}{T_{4a}} = \frac{T_{3s}}{T_3}$$$$T_{3s} = T_2 * \left(\frac{T_3}{T_2}\right)^{\frac{k-1}{k*\eta_c}}$$Where,ηc = 0.89 (Isentropic Efficiency)k = 1.4Given the above values, we can solve for T3s as follows:$$T_{3s} = 572 * \left(\frac{1100}{572}\right)^{\frac{1.4-1}{1.4*0.89}}$$$$T_{3s} = 835.43 K$$On substituting the above values, we get the temperature at the outlet of the turbine (T4) to be 984.44 K.

Next, we need to find the total pressure at the outlet of the turbine (P4) using the following formula:$$\frac{P_4}{P_3} = \left(\frac{T_4}{T_3}\right)^\frac{k}{k-1}$$On substituting the above values, we get the total pressure at the outlet of the turbine (P4) to be 394651.09 Pa.

Therefore, the total temperature after exiting the turbine is 984.44 K, and the total pressure at the turbine exit is 394651.09 Pa.

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