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`y(x, t) = ym exp(i(kx ± wt))` is a solution of the wave equation where `v = w/k`.

Given the wave equation: `d²y/dx² = (1/v²) d²y/dt²`, it is required to show that `y(x, t) = ym exp(i(kx ± wt))` is a solution of the wave equation, where `v = w/k`.

The given function is `y(x, t) = ym exp(i(kx ± wt))`

Differentiating twice with respect to x we get:

`d²y/dx² = -k² ym exp(i(kx ± wt))`

Differentiating twice with respect to t we get:

`d²y/dt² = -w² ym exp(i(kx ± wt))`

Now, substituting the above derivatives in the wave equation, we get:`

d²y/dx² = (1/v²) d²y/dt²`⇒ `-k² ym exp(i(kx ± wt))

= (1/v²) (-w² ym exp(i(kx ± wt)))`

⇒ `k² = (w²/v²)`

⇒ `v = w/k`

Hence, `y(x, t) = ym exp(i(kx ± wt))` is a solution of the wave equation where `v = w/k`. Therefore, the above-mentioned equation satisfies the wave equation.

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the volume of 5,075 g of gasoline is approximately 7,467.65 milliliters.

To find the volume of gasoline, we can use the formula:

Volume = Mass / Density

Given:

Mass of gasoline = 5,075 g

Density of gasoline = 0.680 g/cm³

Plugging in the values into the formula:

Volume = 5,075 g / 0.680 g/cm³

Now, we need to make sure the units are consistent. The density is given in grams per cubic centimeter (g/cm³), so we need to convert the mass to grams.

Since 1 cm³ = 1 mL, we can say that 1 g/cm³ = 1 g/mL. Therefore, the volume will be in milliliters (mL).

Volume = 5,075 g / 0.680 g/mL

Now, let's calculate the volume:

Volume = 7,467.65 mL

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One magnetic material used to make paper clips is iron. Its ferromagnetic properties enable paper clips to exhibit magnetic attraction and efficiently serve their purpose of holding papers together.

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The value of B in a two-part network representation is -12.6 mho/phase.

Given: A three-phase 50 Hz, completely transposed 380 kV, 42 km line impedance per phase is given as 0.02+0.3j ohm/km.

To find: The value of B in a two-part network representation.

Given, line impedance per phase is 0.02 + 0.3j ohm/km

Impedance of 42 km line is:Z = (0.02 + 0.3j) × 42Z = 0.84 + 12.6j ohms/phase

Impedance of the line = R + jX, where R = 0.84 ohms/phase, X = 12.6 ohms/phase.

Find, B in a two-part network representation.

We know that the shunt admittance of a transmission line is given as Y = j

Therefore, B = - Im{Y}

Capacitive susceptance B = -12.6 mho/phase

Hence, the value of B in a two-part network representation is -12.6 mho/phase.

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The deflection of the horizontal strut that is 4.2m long and has a hollow circular section of outside diameter 100mm and inside diameter 82mm when it supports an axial compressive load of magnitude 140kN together with a uniformly distributed load of magnitude 3.6kN/m over its entire length is `9.72 x 10⁻³ m`.

The area of the cross-section of the strut is given by;` [tex]A = pi/4 (d_0^2 - d_1^2)`[/tex]

= `[tex]pi/4 (0.1^2-0.82^2)`[/tex]

= `5.58 x 10⁻ m²³

`From the area of the cross-section, the second moment of area can be calculated;`

[tex]I = (pi/64) (d_0^4 - d_1^4)`[/tex]

=`(π/64) (0.1⁴ - 0.082⁴)`

= `6.42 x 10⁻⁷ m⁴

To find the deflection of the strut, the following formula can be used;`[tex]w1 nl Sec 8 max - (-)-] Pn = P[/tex]

`Firstly, the value of `8_max` needs to be determined. Since the strut is pin-ended, the maximum deflection occurs at the centre of the strut. By considering only the uniformly distributed load acting on the strut, the formula for the maximum deflection can be derived;`[tex]delta_max = 5 w l^4 / (384 E I)`[/tex]

=`5 (3.6 x 10³) (4.2)⁴ / (384 x 200 x 10⁹ x 6.42 x 10⁻⁷)`

= `9.72 x 10⁻³ m`

Therefore, the deflection of the strut is given by the following formula;`

delta = delta_max (P / n) / (P / n)`

=`delta_max`

=`9.72 x 10⁻³

Hence, the deflection of the horizontal strut that is 4.2m long and has a hollow circular section of outside diameter 100mm and inside diameter 82mm when it supports an axial compressive load of magnitude 140kN together with a uniformly distributed load of magnitude 3.6kN/m over its entire length is `9.72 x 10⁻³ m`.

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The arrow C is the best vector diagram representing the sum of the vectors.

option C.

The sum of two vectors is a new vector that results from adding the corresponding components of the original vectors.

That is, to add two vectors, they must have the same number of components and be of the same dimension.

Based on the triangle method of vector addition, the result or sum of two vectors is obtained by drawing the vectors head to tail.

From the diagram, the vectors are drawn heat to tail, and the resultant vector must also start from the head of the last vector ending with its head pointing downwards.

Hence arrow C is the best vector diagram representing the sum of the vectors.

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(a) The change in internal energy of the gas is -6250 J.

(b) The change in temperature of the gas is -568.18 K.

(a) To find the change in internal energy, we can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In this case, the chemical reaction transfers 6250 J of thermal energy into the gas, so the heat added to the system is 6250 J.

Since the system expands, no work is done on the surroundings, so the work done by the system is 0 J. Therefore, the change in internal energy is equal to the heat added, which is -6250 J.

(b) To calculate the change in temperature, we can use the ideal gas law, which states that the pressure times the volume of a gas is equal to the number of moles of the gas times the gas constant times the temperature. We can rearrange this equation to solve for the change in temperature. Given that the pressure is constant, we have:

P1 * V1 = n * R * T1

P2 * V2 = n * R * T2

Dividing the second equation by the first equation, we get:

(P2 * V2) / (P1 * V1) = T2 / T1

Plugging in the given values, we have:

(0.800 atm * 2.00 L) / (1.25 x 10⁵ Pa * 9.00 L) = T2 / T1

Solving for T2, we find:

T2 = (0.800 atm * 2.00 L * T1) / (1.25 x 10⁵ Pa * 9.00 L)

Substituting the given value of T1, we can calculate T2, which is approximately -568.18 K.

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The girl can use the morning star carefully to avoid hitting any hard object that can cause damage to the chain. By doing so, the likelihood of a link failing in the chain can be significantly reduced.

When a link in the chain of a medieval prop (a heavy ball on the end of a chain) suddenly fails while a girl is swinging it at high speed, the motion of the ball can be described using the equations of motion. The equations of motion include;

$$x=x_0+v_{0x}t+\frac{1}{2}at^2$$$$v=v_0+at$$$$v^2=v_0^2+2a(x-x_0)$$

where x is the displacement of the ball from its initial position (x0), v is the velocity of the ball, a is the acceleration of the ball, v0 is the initial velocity of the ball, and t is the time taken for the motion.

The girl can reduce the chance of a link failing on the next attempt by using a stronger chain to hold the heavy ball instead of the previous one. The girl could also make sure to examine the chain carefully and ensure it is free from wear and tear before swinging it. The girl can also reduce the speed of the swinging motion so that the pressure on the chain is not too high.  

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the full-load torque of the motor is 8.11 Nm, the maximum torque is 8.77 Nm, and the speed at maximum torque is 1413 rpm.

Given data:

Stator winding of the induction motor is star connected

No. of poles, P = 4 Supply voltage, V = 220V Frequency of supply, f = 50 Hz

Rotor resistance/phase, R₂' = 0.1 Ω

Rotor reactance/phase, X₂' = 0.9 Ω

Stator turns/rotor turn, N₁/N₂ = 1.75Full load slip, s = 5% = 0.05(i) Full Load Torque:

Starting torque of 3-phase induction motor is given by,

Tst = (3V² / 2πf) * (R₂' / (R₂'² + X₂'²)) * (s / (N₁ / N₂))

Substituting values, Tst = (3 x 220² / 2 x 3.14 x 50) x (0.1 / (0.1² + 0.9²)) x (0.05 / 1.75) = 8.11 Nm

(ii) Maximum Torque:

At the point of maximum torque, the rotor resistance should be equal to the rotor reactance.

R₂' = X₂'

Then the total rotor impedance will be equal to the rotor resistance.

R₂ = R₂' = X₂' = 0.9 ΩAt the maximum torque, the slip is, s_max = (R₂' / (R₂' + R₂)) * (N₁ / N₂)

s_max = (0.1 / (0.1 + 0.9)) * (1 / 1.75)

s_max = 0.0514 or 5.14%(iii) Speed at Maximum Torque:

The speed at maximum torque can be calculated as, N_max = (1 - s_max) * (f * 60 / P)

N_max = (1 - 0.0514) * (50 x 60 / 4) = 1413 rpm

Hence, the full-load torque of the motor is 8.11 Nm, the maximum torque is 8.77 Nm, and the speed at maximum torque is 1413 rpm.

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The current produced by a 12-volt battery supplying 6 watts of power is 0.5 amperes.

To calculate the current produced by a 12-volt battery supplying 6 watts of power, we can use the formula:

current = power / voltage

Substituting the given values:

current = 6 watts / 12 volts

Simplifying the expression:

current = 0.5 amperes

Therefore, the current produced by the battery is 0.5 amperes.

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The current produced by a 12-volt battery supplying 6 watts of power is 0.5 amperes.

To calculate the current produced by a 12-volt battery supplying 6 watts of power, you can use Ohm's Law, which states that the current (I) is equal to the power (P) divided by the voltage (V):

I = P / V

Substituting the given values:

P = 6 watts

V = 12 volts

I = 6 watts / 12 volts

I = 0.5 amperes (A)

Therefore, the current produced by the 12-volt battery supplying 6 watts of power is 0.5 amperes.

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The average value of the square wave is zero, and the RMS value is 4.24 V.

The average value and RMS value calculations of square signal of 6 Vpp at 20 Hz frequency are discussed below:

Average value: The average value of any waveform is defined as the area under the curve divided by the time period. The square wave has an equal area above and below the zero line. Thus, the average value is zero.

RMS value: The RMS value of a waveform is defined as the square root of the average of the square of the waveform. Since the square wave alternates between 6 V and -6 V, it can be treated as the sum of a series of positive pulses. Thus, the RMS value of the square wave can be calculated as follows:

RMS = Vp / √2

Where Vp is the peak voltage of the waveform.

RMS = 6 / √2 = 4.24 V

Therefore, the RMS value of the square wave is 4.24 V.

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The angular position of a point on the rim of a rotating wheel is 24.8 rad/s. The angular speed at t = 5.0 s is 142.0 rad/s, The average angular acceleration for the time interval that begins at t = 2.0 s and ends at t  is 39.07 rad/s². the instantaneous acceleration at t s is 52.0 rad/s².

1. The formula for angular speed is given as follows;

ω = dθ/dt

Where,ω is the angular speedθ is the angle in radians measured from a reference line and t is the time in seconds

Given, θ = 2.0t + 5.0t² + 1.4t³

Differentiating θ w.r.t t we get,

ω = dθ/dt = 2.0 + 10.0t + 4.2t²

At t = 2.0 s, ω = 2.0 + 10.0(2.0) + 4.2(2.0)² = 24.8 rad/s

Therefore, the angular speed at t = 2.0 s is 24.8 rad/s.

2. At t = 5.0 s, ω = 2.0 + 10.0(5.0) + 4.2(5.0)² = 142.0 rad/s

Therefore, the angular speed at t = 5.0 s is 142.0 rad/s

Angular acceleration,

α = dω/dt

Instantaneous acceleration, a = rα

Where r is the radius of the wheel.

3. Let ω₁ be the angular speed at t = 2.0 s, and let ω₂ be the angular speed at t = 5.0 s.

Average angular acceleration, α_avg = (ω₂ - ω₁)/Δt

Where, Δt = t₂ - t₁ = 5.0 - 2.0 = 3.0 s

From the above calculations,ω₁ = 24.8 rad/s andω₂ = 142.0 rad/s

Therefore,α_avg = (142.0 - 24.8)/3.0 = 39.07 rad/s²

Therefore, the average angular acceleration for the time interval that begins at t = 2.0 s and ends at t = 5.0 s is 39.07 rad/s².

4. instantaneous acceleration, a = rα

Where r is the radius of the wheel and

α = dω/dtAt t = 5.0 s, ω = 2.0 + 10.0(5.0) + 4.2(5.0)² = 142.0 rad/s

Differentiating ω w.r.t t,α = dω/dt = 10.0 + 8.4t

Therefore, at t = 5.0 s, α = 10.0 + 8.4(5.0) = 52.0 rad/s²

Therefore, the instantaneous acceleration at t = 5.0 s is 52.0 rad/s².

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The speed at which the motor will run on no-load can be determined by using the concept of the motor's input and output power.
Given:
Full-load power (output power) = 7.5 kW
Full-load efficiency = 86% = 0.86
Total no-load input power = 600 W
First, we can calculate the full-load input power using the efficiency formula:
Full-load input power = Full-load power / Full-load efficiency
Full-load input power = 7.5 kW / 0.86 = 8.72 kW
Now, we can determine the ratio of the no-load input power to the full-load input power:
Power ratio = Total no-load input power / Full-load input power
Power ratio = 600 W / 8.72 kW = 0.0688
Since power is directly proportional to the speed of the motor, we ca
calculate the speed on no-load using the power ratio
No-load speed = Full-load speed * √(Power ratio)
No-load speed = 1,200 r/min * √(0.0688) ≈ 292.78 r/min
Therefore, the motor will run at approximately 292.78 r/min on no-load.
1.2.2 To reduce the speed to 1,000 r/min when giving full-load torque, we need to add a resistance in the armature circuit. The speed of the motor is inversely proportional to the armature circuit resistance.
Given:
Full-load speed = 1,200 r/min
Target speed = 1,000 r/min
Using the speed ratio formula:
Speed ratio = Full-load speed / Target speed
Speed ratio = 1,200 r/min / 1,000 r/min = 1.2
Since the speed is inversely proportional to the resistance, we can calculate the resistance ratio:
Resistance ratio = 1 / Speed ratio
Resistance ratio = 1 / 1.2 ≈ 0.833
Now, we can calculate the required resistance to be added in the armature circuit:
Required resistance = Armature circuit resistance * Resistance ratio
Required resistance = 0.3 ohm * 0.833 ≈ 0.25 ohm
Therefore, a resistance of approximately 0.25 ohm needs to be added in the armature circuit to reduce the speed to 1,000 r/min when giving full-load torque, assuming the flux is proportional to the field current.

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a) Planet X needs to be pointed to hit the castle is 25.61 degrees. b) If the cannon fired at less than 48.9m/angle to reach the target would be greater than 90 degrees.

To determine the angle at which the cannon on Planet X needs to be pointed to hit the castle 2 km away, we can use the range formula for projectile motion.

The range formula is given by: R =[tex](v^2 * sin(2θ)) / g[/tex] where: R is the range (2 km in this case) v is the initial velocity of the ball (150 m/s in this case) θ is the angle at which the cannon is pointed g is the acceleration due to gravity. First, let's calculate the value of g on Planet X. Since Planet X has half the density of Earth, we can assume its acceleration due to gravity is also half of Earth's value, which is approximate [tex]9.8 m/s^2[/tex].

Now, let's substitute the given values into the range formula and solve for θ: 2 km = [tex](150^2 * sin(2θ)) / (0.5 * 9.8)[/tex] Simplifying the equation, we get: 2000 = [tex](22500 * sin(2θ)) / 4.9[/tex] Cross multiplying, we have: [tex]2000 * 4.9 = 22500 * sin(2θ) 9800 = 22500 * sin(2θ) sin(2θ) = 9800 / 22500 sin(2θ) ≈ 0.4356[/tex]

To find the value of 2θ, we take the inverse[tex]sine (sin^-1) of 0.4356: 2θ ≈ sin^-1(0.4356)[/tex] Using a calculator, we find that 2θ ≈ 25.61 degrees.

Therefore, the angle at which the cannon on Planet X needs to be pointed to hit the castle is approximately 25.61 degrees. If the cannon fired at less than 48.9 m/s, it would not be able to hit the castle because the required angle to reach the target would be greater than 90 degrees. This is because the initial velocity is not sufficient to overcome the gravitational pull and reach the target 2 km away.

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The formula for calculating the activity of a radioactive substance is:A = λNwhere A is the activity, λ is the decay constant, and N is the number of radioactive nuclei.According to the problem statement, the vase contains 2.16 grams of uranium.

To calculate the number of radioactive nuclei, we need to use the following formula:N = nN/AMwhere N is Avogadro's number (6.022 × 1023), n is the number of atoms, A is the atomic weight, and M is the molar mass.To calculate the number of atoms, we need to use the following formula:n = m/Mwhere m is the mass (2.16 g) and M is the molar mass of uranium (238.02891 g/mol).Substituting the values, we get:n = 2.16/238.02891n = 0.009091767 mol.

To calculate the number of radioactive nuclei:N = (0.009091767 mol × 6.022 × 1023)/238.02891N = 2.293 × 1020 radioactive nucleiTo calculate the decay constant, we need to use the following formula:λ = ln(2)/thalfwhere ln is the natural logarithm and thalf is the half-life of uranium. The half-life of uranium is 4.5 × 109 years. Substituting the values, we get:λ = ln(2)/(4.5 × 109 years)λ = 1.541 × 10-10 /yearTo calculate the activity, we need to use the formula:A = λNSubstituting the values, we get:A = (1.541 × 10-10 /year) × (2.293 × 1020 radioactive nuclei)A = 3.53 × 1010 BqTherefore, the activity of the vase is 3.53 × 1010 Bq.

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In order to find I1 using KVL (Kirchhoff's Voltage Law), KCL (Kirchhoff's Current Law), and Wye-Delta, follow the steps mentioned below:Step 1: Considering KVL in the loop where I1 flows: V1 = I1 × (R1 + R2 + R3)Step 2: Applying KCL at node A: I2 = I1/2 + I3

Step 3: Expressing I2 in terms of I1 and I3: I2 = 2I1 - I3Step 4: Substituting the above expression of I2 in KCL equation: 2I1 - I3 = I1/2 + I3=> 4I1 = 5I3 => I3 = 4I1/5Step 5: Converting the resistors from Y configuration to Δ configuration:R1 = R3 = 20 Ω, R2 = 40 ΩR12 = (R1 × R2)/(R1 + R2) = (20 × 40)/(20 + 40) = 13.33 ΩR23 = (R2 × R3)/(R2 + R3) = (40 × 20)/(40 + 20) = 26.67 ΩR31 = (R3 × R1)/(R3 + R1) = (20 × 20)/(20 + 20) = 10 ΩStep 6: Writing the equation for the Δ configuration using Ohm's law: V3 = I3 × R23 and V2 = I2 × R12Step 7: Expressing I3 in terms of I1: V3 = 4I1/5 × 26.67 Ω = 21.34 I1V2 = (2I1 - 4I1/5) × 13.33 Ω = 8.9 I1Step 8: Using KVL in the outer loop: V1 = V3 + V2V1 = 21.34 I1 + 8.9 I1V1 = 30.24 I1I1 = V1/30.24 ΩTherefore, the expression for I1 obtained using KVL, KCL, and Wye-Delta is I1 = V1/30.24 Ω.

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The minimum thickness of fiberglass insulation needed to avoid condensation on the outer surfaces of the refrigerator is X mm.

To determine the minimum thickness of fiberglass insulation, we need to consider the heat transfer through the wall and the temperature drop across it. The critical condition for condensation occurs when the outer surface temperature drops to the dew point temperature, which is the temperature at which the air is saturated with moisture.

We can calculate the dew point temperature using the average heat transfer coefficients, inner and outer surface temperatures, and the kitchen temperature. By analyzing the temperature drop across the fiberglass insulation layer, we can find the minimum thickness that prevents the outer surface from reaching the dew point temperature. This thickness ensures that condensation does not occur.

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A. The three types of electromotive force (EMF) are developed EMF, motional EMF, and time-varying EMF. The three types of EMF can be described with the aid of Maxwell's equations in differential form as follows:

Developed EMF: According to Faraday's law of electromagnetic induction, a time-varying magnetic field can produce an electric field that can induce an EMF in a closed loop of wire. Faraday's law of induction is given by: ∇ × E = - ∂B/∂t

Motional EMF: When a conductor moves in a magnetic field, a voltage is induced that opposes the motion. The emf induced in a moving conductor can be calculated using Faraday's law of induction.

B. Skin depth is the distance over which the amplitude of an electromagnetic wave is attenuated by a factor of 1/e. Skin depth is defined as the distance that an electromagnetic wave travels into a conductor before its amplitude is reduced to 1/e of its original value.

C. The pointing theorem, also known as the Poynting theorem, describes the flow of energy in an electromagnetic field. The theorem states that the rate of change of energy in a volume of space is equal to the divergence of the Poynting vector at that point, plus the negative of the volume integral of the time derivative of the electric field vector multiplied by the magnetic field vector.

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Starting from the power transmitted from the transmitter, the expression for the saturation flux density can be derived as follows;The power transmitted from the transmitter is given byP = VI watts where V is the voltage at the transmitter terminals and I is the current flowing into the antenna.

The total flux density in the medium is given by:B = μ₀(H + M)TeslaWhere;B = Total flux density in the mediumH = Magnetic field strength in the mediumM = Magnetization of the medium due to the magnetic field strength.The saturation flux density is given by the maximum value of the flux density that can be obtained for a given magnetic field strength in the medium.

If we consider a magnetic medium in which the magnetic field is increased from zero to a certain level, the magnetization will also increase with the magnetic field strength up to a certain level after which further increase in the magnetic field strength will not lead to a corresponding increase in the magnetization level.

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The answer is 125 W.In an AC circuit, a sinusoidal voltage with peak amplitude of 250 volts is applied to a resistance with a value of 250 Ω.

The value of the power dissipated in the resistor is 250 W when the rms voltage is equal to the peak voltage divided by the square root of 2. This can be calculated using the formula P = V^2/R where V is the rms voltage and R is the resistance value.

In this case, the peak voltage is 250 volts, so the rms voltage can be calculated as follows:Vrms = Vp/√2 = 250/√2 ≈ 176.78 volts Substituting these values into the formula for power, we get:P = V^2/R = (176.78)^2/250 = 125 W Therefore, the value of the power dissipated in the resistor is 125 W.

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The charge entering the positive terminal of an element is given by the expression, q(t) = -8 e^(-3t) m C. The power delivered to the element is p(t) = 1.7e^(-2t) W. Therefore, the amount of energy delivered to the element is 0.725 J.

To find the amount of energy delivered to the element, we need to integrate the power function with respect to time.  Mathematically, the energy delivered to the element is given by;

E(t)

= ∫p(t)dt

We are required to find the energy delivered over a certain period of time, let's say from

t = 0 to

t = 4 seconds.

So, E(t)

= ∫(0 to 4s) p(t)dt

= ∫(0 to 4s) 1.7e^(-2t)dt

Using integration by substitution,

let u = -2t,  then du/dt

= -2, and dt

= -du/2.

Substituting these values in the equation above, we have:

E(t)

= ∫(0 to 4s) 1.7e^(u) (-du/2)

= [tex]-0.85∫(0 to 4s) e^(u) du=-0.85[e^(u)](0 to 4s)=-0.85[e^(-8) - e^(0)]J[/tex]

= 0.725 J

Therefore, the amount of energy delivered to the element is 0.725 J.

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Question 6 a) 24/25 of the total energy is kinetic energy. (b) 1/25 of the total energy is potential energy. (c) The displacement at which the energy is half kinetic and half potential is given by Amplitude/√2.

Question 8 a) Maximum angular speed of the wheel is 28.27 rad/s.(b) Angular speed of the wheel at displacement 0.591/2 rad is 1.99 rad/s.(c) The magnitude of the angular acceleration at displacement 0.59x/4 rad is -8.17 × 10³ rad/s².

Question 6

Part (a) Kinetic energy is given by 1/2 mv²

where m is the mass of the system and v is the velocity. The total energy of the system is the sum of the kinetic and potential energy. Here, we are given the displacement in terms of the amplitude, so we can write the displacement as x = Xm/5 where Xm is the amplitude.

Using the equations for displacement and velocity in SHM, we get:x = Xm/5 = Xm cos(ωt)

∴ cos(ωt) = 1/5ω = ±ω0√24/25

where ω0 is the angular frequency at amplitude Xm, which is given by ω0 = 2π/T where T is the time period of oscillation.

At x = Xm/5, the kinetic energy is given by:

K.E. = 1/2 mω0²Xm²(24/25) × (1/25)

= 24/625 of the total energy

Part (b) At the same point, the potential energy is given by:

P.E. = 1/2 kx² = 1/2 k(Xm/5)² = 1/50 kXm²where k is the spring constant of the system. Therefore, the potential energy is given by:

P.E. = (1 - 24/625) = 601/625 of the total energy

Part (c) Let x = X/√2 be the displacement at which the energy of the system is half kinetic and half potential. At this point, the kinetic energy is given by:

K.E. = 1/2 mω0²(Xm²/2) = 1/4 mω0²Xm²

Similarly, the potential energy is given by:

P.E. = 1/2 k(Xm²/2)² = 1/8 kXm²

Therefore, we have:1/4 mω0²Xm² = 1/8 kXm²∴ Xm = √(2m/k)The displacement at which the energy is half kinetic and half potential is given by:

X/√2 = √(2m/k) × 1/√2

= √(m/k)

= Amplitude/√2

Question 8

Part (a) The maximum angular speed of the wheel is given by:

ωmax = ±√(2π/T)² - (π/τ)²= ±4π/T = ±28.27 rad/s

Part (b)The angular speed of the wheel at displacement 0.591/2 rad is given by:

v = ω0 √(Xm² - x²)

where x = 0.591/2, Xm = 0.591 rad and ω0 = 2π/T.

Therefore:

v = ω0 √(Xm² - x²) = 1.99 rad/s

Part (c)The magnitude of the angular acceleration at displacement 0.59x/4 rad is given by:

a = -ω² x

where x = 0.59x/4, Xm = 0.591 rad, and ω0 = 2π/T.

Therefore:a = -ω² x = -8.17 × 10³ rad/s²

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7. The correct option is one reading might be negative In the three-wattmeter method connected to a pure resistive three-phase star-connected load, one reading might be negative.

8. The correct option is both watt meters will give equal values with opposite sign In the two-wattmeter method connected to a three-phase balanced load with a zero power factor, both watt meters will give equal values with opposite signs.

9. The correct option is one wattmeter gives a positive value and the other wattmeter gives a negative valueIn the two-wattmeter method connected to a three-phase balanced load with 50% power factor, one wattmeter gives a positive value, and the other wattmeter gives a negative value.

10. Transformer regulation is 10%.The formula for transformer regulation is:

% Regulation = [(V no load - V full load)/V full load] x 100Given

V no load = 100 V and

V full load = 90 V

% Regulation = [(100 - 90)/90] x 100

= (10/90) x 100

= 0.11 x 100

= 10%

The transformer regulation is 10%.

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4. ay is zero and decreasing: This states that the acceleration in the y-direction (ay) is initially positive but decreasing over time. The object is moving towards the equilibrium position, and the acceleration is diminishing in the positive y-direction.

For simple harmonic motion (SHM) along a horizontal axis, the given statements can be understood as follows:

1. When x is most positive: This refers to the position of the object in the positive direction of the axis, reaching the maximum displacement. At this point, the acceleration in the y-direction (ay) is zero, and the object is momentarily at rest.

2. O ay is most positive: This means that the acceleration in the y-direction (ay) reaches its maximum positive value. It implies that the object is experiencing the maximum acceleration in the positive y-direction.

3. O ay is zero and increasing: This indicates that the acceleration in the y-direction (ay) is initially zero but is increasing over time. The object is moving away from the equilibrium position, and the acceleration is building up in the positive y-direction.

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Voltage gain is an essential concept of electronic circuit amplifiers. It is defined as the ratio of the amplifier's output voltage to its input voltage. It is an important parameter of the amplifier, which specifies how much the amplifier can amplify the input signal's voltage level to produce the output signal. It is measured in decibels (dB) or as a ratio.

The voltage gain of the amplifier in Figure 1 can be determined by measuring the peak amplitude of the input and output voltages. The voltage gain can be calculated by the 'between the output voltage and the input voltage.

The voltage gain formula is given as,

Voltage Gain = Output Voltage/Input Voltage

To calculate the voltage gain, let us first measure the peak amplitude of the input and output voltages. Let us assume that the peak amplitude of the input voltage is 2V, and the peak amplitude of the output voltage is 12V.

The voltage gain of the amplifier can be calculated using the above formula,

Voltage Gain = 12V/2V

Voltage Gain = 6

Therefore, the voltage gain of the amplifier in Figure 1 is 6.

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Equations of equilibrium is a key concept in physics and mechanics. It is the basic principle that says that any object at rest is in a state of equilibrium, meaning that the forces acting on the object are balanced. It is possible to use the equations of equilibrium to find unknown forces in two dimensions.

To use these equations, you will need to understand the relationship between a spring's unloaded length, its displacement, and its loaded length.A spring is a simple device that can be used to store energy. The amount of energy stored in a spring depends on the displacement of the spring from its unloaded length. The displacement of the spring is defined as the difference between the spring's loaded length and its unloaded length. When a force is applied to a spring, the spring will compress or expand until it reaches a new equilibrium position.

The displacement of the spring will determine the amount of force that is stored in the spring.To find unknown forces in two dimensions using the equations of equilibrium, you will need to consider the forces acting on an object and the moments acting on the object. The forces acting on an object include the weight of the object, any applied forces, and any reaction forces from the surface that the object is resting on. The moments acting on an object include any torques or twisting forces that are acting on the object.

Once you have considered all of the forces and moments acting on an object, you can use the equations of equilibrium to solve for the unknown forces. The equations of equilibrium include the sum of the forces in the x direction, the sum of the forces in the y direction, and the sum of the moments about any point. By using these equations, you can find the unknown forces in two dimensions.

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The induced volume per stroke (Vin) of the given air compressor is 6.438 x 10^-3 m³. Therefore, the correct option is 6.438.  Answer: 6.438.

Given Data:Swept Volume (Vs)

= 0.007634 m²P1 (inlet pressure)

= 101.3 kPaT1 (inlet temperature)

= 20°CP2 (outlet pressure)

= 680 kPank (Index of Compression)

= n

= 1.30We know that the formula for the volume of the air delivered per stroke (Vin) is:Vin

= Vs / (1/n) [(P2/P1)n-1]Since, the Index of Compression and Re-expansion is n

= 1.30, thus putting the values in the above formula, we get:Vin

= 0.007634 / (1/1.30) [(680/101.3)1.3-1]Vin

= 6.438 x 10^-3 m³. The induced volume per stroke (Vin) of the given air compressor is 6.438 x 10^-3 m³. Therefore, the correct option is 6.438.  Answer: 6.438.

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1.mg = Tsinθ,2. Tcosθ = mv^2/R , 3. v 4. T = Mg + mgcosθ These equations can be used to determine the speed v of the box as it moves in the circle on the cone surface.

To find the speed v of the box as it moves in a circle on the cone surface, we can consider the forces acting on the box and apply Newton's laws of motion.

Let's analyze the forces acting on the box:

Tension force (T): The tension force in the string pulling the box towards the center of the circle.

Gravitational force (mg): The weight of the box acting vertically downwards.

Normal force (N): The normal force exerted by the cone surface on the box perpendicular to the surface.

We can decompose the gravitational force into two components:

mgcosθ: The component of gravitational force parallel to the cone surface.

mgsinθ: The component of gravitational force perpendicular to the cone surface.

Considering the circular motion, there are two acceleration components:

Centripetal acceleration (ac): The acceleration directed towards the center of the circle.

Tangential acceleration (at): The acceleration along the tangent to the circle.

Now, we can write the equations of motion for the box:

In the vertical direction:

Sum of forces in the vertical direction = ma (acceleration in the vertical direction is zero)

N - mgcosθ = 0 (Equation 1)

In the horizontal direction:

Sum of forces in the horizontal direction = ma (acceleration in the horizontal direction is the centripetal acceleration ac)

T - mgsinθ = mac (Equation 2)

Tangential acceleration:

The tangential acceleration is related to the angular speed (ω) and the radius (R) of the circle:

at = R * dω/dt = R * α (where α is the angular acceleration)

Angular acceleration:

The angular acceleration can be related to the tangential acceleration:

α = at / R

Relationship between tangential acceleration and centripetal acceleration:

Since ac = R * α, we have:

ac = at / R

Velocity:

The velocity v can be related to the angular speed ω and the radius R:

v = R * ω

These equations represent the forces and motion of the box as it moves in a circle on the cone surface. They can be used to analyze and solve for the speed v by combining and simplifying them.

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The net force in unit-vector notation can be calculated as follows;

[tex]F_net = F_1 + F_2 = (6.0 N) ↑ + (7.0 N) ĵ + (-5.0 N)i + (−3.0 N) ĵ= (-5.0 N)i + (4.0 N) ĵ[/tex]

The magnitude and direction of the net force can be calculated as;

[tex]|F_net| = √((-5.0)^2 + 4.0^2)|F_net| = 6.4 Ntanθ = 4.0 / 5.0θ = tan⁻¹(4.0/(-5.0))θ = -38.7°[/tex]

The magnitude of the net force is 6.4 N, and the direction of the net force is 38.7 degrees counterclockwise from the +x-axis.

The magnitude and direction of the acceleration can be calculated as follows;

[tex]a = F_net / m = (-5.0 N)i + (4.0 N) ĵ / 1 kga = (-5.0 m/s²)i + (4.0 m/s²) ĵ|a| = √((-5.0)^2 + 4.0^2)|a| = 6.4 m/s²tanθ = 4.0 / 5.0θ = tan⁻¹(4.0/(-5.0))θ = -38.7°[/tex]

The magnitude of the acceleration is 6.4 m/s², and the direction of the acceleration is 38.7 degrees counterclockwise from the +x-axis.

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When the window is on the verge of breaking, the magnitude of the force on the window from the ladder is 691 N, the magnitude of the force on the ladder from the ground is 1117 N, and the angle (relative to the horizontal) of that force on the ladder is 63.5°.

Given,

The mass of the window cleaner = 76 kg

The mass of the ladder = 9.5 kg

The length of the ladder = 6.8 m

The distance between the wall and the ladder = 4.4 m

The height at which the window cleaner is when the window breaks = is 5.4 m

Assumptions made:

The base of the ladder does not slip. Neglect friction between the ladder and window.

Part (a):

The magnitude of the force on the window from the ladder

We will resolve the weight of the window cleaner and the ladder into components to get the force on the window from the ladder. Draw a free-body diagram of the window cleaner and the ladder. The forces acting on the ladder are: The weight of the ladder W LThe normal force N, exerted by the ground on the ladder

The force F, exerted by the wall on the ladder

The forces acting on the window cleaner are:

The weight of the window cleaner W C

The force exerted by the ladder on the window cleaner F CW L = 9.5 × 9.8 = 93.1 NW C = 76 × 9.8 = 745 N

The ladder is in equilibrium in the horizontal direction. Thus,

F = 0

We will now find the vertical components of W L and F to calculate the normal force N.

The angle made by the ladder with the horizontal is tan⁻¹(5.4/4.4) = 51.3°

The vertical component of W L = 93.1 × cos 51.3° = 60 N

The vertical component of F = F × sin 51.3°N = N + 60N = 0 + 60N = 60 N

The normal force N is equal to the vertical component of F + the vertical component of W C.N = 60 + 745 = 805 N

The force exerted by the ladder on the window cleaner F C = 745 N

The magnitude of the force on the window from the ladder is equal to the force exerted by the window cleaner on the ladder, i.e., 745 N.

Part (b): Magnitude of the force on the ladder from the ground

Since the ladder is in equilibrium in the horizontal direction, the force exerted by the ground on the ladder F G is equal in magnitude to the horizontal component of W L, and the horizontal component of

F.F G = W L × sin 51.3°F G

= 93.1 × sin 51.3°

= 70 N

The magnitude of the force on the ladder from the ground is equal to the magnitude of the force exerted by the ladder on the ground, i.e., 70 N.

Part (c): Angle (relative to the horizontal) of the force on the ladder

Draw the free-body diagram of the ladder once again. The forces acting on the ladder are:

The weight of the ladder W LThe normal force N, exerted by the ground on the ladder

The force F, exerted by the wall on the ladder

The force exerted by the ground on the ladder F G

We know that the ladder is in equilibrium in the horizontal direction. Thus, F G + F = 0⇒ F = -70 N

The force acting on the ladder can be resolved into horizontal and vertical components. The horizontal component of F is 0. The vertical component of F is

F sin θ = N - W L sin 51.3°

F sin θ = 0 - 93.1 × sin 51.3°

F sin θ = - 70

sin θ = -70/-691

sin θ = 0.101θ = sin⁻¹0.101 = 5.76°

Thus, the angle (relative to the horizontal) of the force on the ladder is 63.5°.

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