In a phasor representation of a transverse wave on a string, what does the length of the phasor represent?

Answer:

The factor of the diameter is 0.95.

Explanation:

Given that,

Power of old light bulb = 54.3 W

Power = 60 W

We know that,

The resistance is inversely proportional to the diameter.

[tex]R\propto\dfrac{1}{D}[/tex]

The power is inversely proportional to the resistance.

[tex]P\propto\dfrac{1}{R}[/tex]

[tex]P\propto D^2[/tex]

We need to calculate the factor of the diameter of the filament reduced

Using relation of power and diameter

[tex]\dfrac{P_{i}}{P_{f}}=\dfrac{D_{i}^2}{D_{f}^2}[/tex]

Put the value into the formula

[tex]\dfrac{D_{i}^2}{D_{f}^2}=\dfrac{54.3}{60}[/tex]

[tex]\dfrac{D_{i}}{D_{f}}=0.95[/tex]

[tex]D_{i}=0.95 D_{f}[/tex]

Hence, The factor of the diameter is 0.95.

Answer:

Explanation:

Po = 60 W

P = 54.3 W

Let the initial diameter of the filament is do and the final diameter of the filament is d.

Let the voltage is V and the initial resistance is Ro and the final resistance is R.

The formula for power is given by

P = V²/R

The resistance of the filament is inversely proportional to the square of the diameter of the filament. As voltage is constant so the power is

Power α diameter²

So, initial power is

Po α do²     ..... (1)

Final power is

P α d²         ..... (2)

Divide equation (2) by equation (1), we get

P / Po = d² / do²

54.3 / 60 = d² / do²

d² / do² = 0.905

d = 0.95 d

Thus, the diameter of the filament is reduced to a factor of 0.95 .  

Answer:

a) P = 2319.6[kPa]; b) 2.6%

Explanation:

Since the problem data is not complete, the following information is entered:

A 1.78-m3 rigid tank contains steam at 220°C. One-third of the volume is in the liquid phase and the  rest is in the vapor form. Determine (a) the pressure of the steam, and (b) the quality of the saturated  mixture.

From the information provided in the problem we can say that you have a mixture of liquid and steam.

a) Using the steam tables we can see (attached image) that the saturation pressure at 220 °C is equal to:

[tex]P_{sat} =2319.6[kPa][/tex]

[tex]v_{f}=0.001190[m^{3}/hr]\\v_{g}=0.08609[m^{3}/hr]\\[/tex]

b) Since the specific volume of the gas and liquid is known, we can find the mass of each phase using the following equation:

[tex]m_{f}=\frac{V_{f} }{v_{f} } \\m_{g}=\frac{V_{g} }{v_{g} } \\where:\\V_{f}=volume of the fluid[m^3]\\v_{f}=specific volume of the fluid [m^3/kg]\\[/tex]

We know that the volume of the fluid is equal to:

[tex]V_{f}=1/3*V_{total} \\V_{total}=1.78[m^3]\\[/tex]

Now we can find the mass of the gas and the liquid.

[tex]m_{f}=\frac{1/3*1.78}{0.001190} \\m_{f}=498.6[kg]\\m_{g}=\frac{2/3*1.78}{0.08609}\\m_{g}=\ 13.78[kg][/tex]

The total mass is the sum of both

[tex]m_{total} =m_{g} + m_{fluid} \\m_{total} = 498.6 + 13.78\\m_{total} = 512.38[kg][/tex]

The quality will be equal to:

[tex]x = \frac{m_{g} }{m_{T} }\\ x= \frac{13.78}{512.38} \\x = 0.026 = 2.6%[/tex]

Answer:

75.5g

Explanation:

From the ionic equation, we can write

[tex]CU^{2+}+SO^{2-}_{4}\\[/tex]

next we find the number of charge

Note Q=it

for i=8.5A, t=3.75 to secs 3.75*60*60=13500secs

hence

[tex]Q=8.5*13500\\Q=114750C[/tex]

Since one faraday represent one mole of electron which equal 96500C

Hence the number of mole produced by 114750C is

114750/96500=1.2mol

The mass of copper produced is

[tex]mol=\frac{mass}{molar mass} \\mass=mole*molar mass\\mass=1.2*63.5\\mass=75.5g[/tex]

Hence the amount of copper produced is 75.5g

Answer:

75.5 g.

Explanation:

Dissociation equation:

Cu2+ + SO4^2- --> CuSO4

Q = I * t

Where,

I = current

= 8.5 A

t = time

= 3.75 hours

= 13500 s

Q = 8.5 * 13500

= 114750 C

1 faraday represent one mole of electron which equal 96500C

Number of mole of Cu2+

= 114750/96500

= 1.19 mol.

Mass = number of moles * molar mass

= 1.19 * 63.5

= 75.5 g.

Answer:

3.76 m/s

Explanation:

Instantaneous velocity: This can be defined as the velocity of an object in a non uniform motion. The S.I unit is m/s.

v' = dx(t)/dt..................... Equation 1

Where v' = instantaneous velocity, x = distance, t = time.

Given the expression,

x(t) = 28.0 m + (12.4 m/s)t - (0.0450 m/s³)t³

x(t) = 28 + 12.4t - 0.0450t³

Differentiating x(t) with respect to t.

dx(t)/dt = 12.4 - 0.135t²

dx(t)/dt = 12.4 - 0.135t²

When t = 8.00 s.

dx(t)/dt = 12.4 - 0.135(8)²

dx(t)/dt = 12.4 - 8.64

dx(t)/dt = 3.76 m/s.

Therefore,

v' = 3.76 m/s.

Hence, the instantaneous velocity = 3.76 m/s

Final answer:

To find the bird's instantaneous velocity at t = 8.00s, we differentiate its position function to get v(t) = 12.4 m/s - 0.135 m/s^2 × t^2, then substitute t = 8.00s to find v(8.00) = 3.76 m/s east.

Explanation:

The question asks for the instantaneous velocity of a bird flying due east when t = 8.00s, given the position function x(t) = 28.0 m + (12.4 m/s)t – (0.0450 m/s3)t3. To find the instantaneous velocity, we need to differentiate the position function with respect to time (t) to get the velocity function, v(t).

First, let's differentiate x(t):

Derivative of 28.0 m is 0 since it's a constant.

Derivative of (12.4 m/s)t is 12.4 m/s, as the derivative of t is 1.

Derivative of (-0.0450 m/s3)t3 is -0.135 m/s2 × t2, using the power rule for derivatives.

So, the velocity function is v(t) = 12.4 m/s - 0.135 m/s2 × t2. To find the instantaneous velocity at t = 8.00s, we plug in t = 8.00 into the velocity function:

v(8.00) = 12.4 m/s - 0.135 m/s2 × (8.002)

Calculating this gives us:

v(8.00) = 12.4 m/s - 0.135 m/s2 × 64.00 = 12.4 m/s - 8.64 m/s = 3.76 m/s

Therefore, the instantaneous velocity of the bird when t = 8.00s is 3.76 m/s east.

Answer:

(a) R = 1Ω

(b) τ = 3

Explanation:

The time constants of the given circuits are as follows

[tex]\tau_{RL} = \frac{L}{R}\\\tau_{RC} = RC[/tex]

If the two circuits have equal time constants, then

[tex]\frac{L}{R} = RC\\R^2 = \frac{L}{C} = \frac{3}{3} = 1\\R = 1\Omega[/tex]

Therefore, the time constant in any of the circuits is

[tex]\tau = RC = 3[/tex]

(a). Value of resistance R is  1 ohm.

(b). The Value of time constant will be 3 second.

The time constant is defined as, time taken by the system to reach at 63.2% of its final value.

Time constant in RL circuit is,  [tex]=\frac{L}{R}[/tex]

Time constant in RC circuit is, [tex]=RC[/tex]

Since, in question given that both circuit have same time constant.

So,    [tex]RC=\frac{L}{R}\\\\R^{2}=\frac{L}{C}\\\\R=\sqrt{\frac{L}{C} }[/tex]

substituting L = 3H and C = 3F in above expression.

         [tex]R=\sqrt{\frac{3}{3} }=1ohm[/tex]

Time constant = [tex]RC=1*3=3s[/tex]

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Answer:

The potential at point P is -111 Volt.

Explanation:

Given that,

Distance = 0.033 m

Work = 111 ev

Suppose what is the potential at point P?

We need to calculate the potential at point P

Using formula of potential

[tex]W=qV[/tex]

[tex]V=\dfrac{W}{q}[/tex]

Where, W = work

q = charge

Put the value into the formula

[tex]V= \dfrac{111\times1.6\times10^{-19}}{-1.6\times10^{-19}}[/tex]

[tex]V=-111\ V[/tex]

Hence, The potential at point P is -111 Volt.

The question is about the process of bringing an electron from rest to rest near a fixed charge q at a specific distance, and the energy required for this process. The answer explains the equation for the energy required per unit charge and the concept of electron volts (eV) as an energy unit.

The question is asking about the process of bringing an electron from rest at an infinite distance to rest at a point located a distance of 0.033 m from a fixed charge q using an external agent. The process requires 111 eV of energy to perform the necessary work. The equation for the energy required per unit charge is given, which relates the potential energy to the charge and distance. The concept of electron volts (eV) as an energy unit is also mentioned.

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Answer:

Explanation:

Given

Gauge Pressure of a tank is

[tex]P_{gauge}=55\ kPa[/tex]

at that place atmospheric Pressure is  [tex]h=72.1\ cm\ of\ Hg[/tex]

Density of mercury [tex]\rho _{Hg}=13600\ kg/m^3[/tex]

Atmospheric Pressure in kPa is given by

[tex]P_{atm}=\rho _{Hg}\times g\times h[/tex]  

[tex]P_{atm}=13600\times 9.8\times 0.721[/tex]

[tex]P_{atm}=96.09\ kPa[/tex]

and Absolute Pressure is summation of gauge pressure and atmospheric Pressure

[tex]P_{abs}=P_{gauge}+P_{atm}[/tex]

[tex]P_{abs}=55+96.09=151.09\ kPa[/tex]                        

Answer:

The fraction of collision is [tex]4.00\times10^{-15}[/tex]

Explanation:

Given that,

Temperature = 47°C

Activation energy = 88.20 KJ/mol

From Arrhenius equation,

[tex]k=Ae^{-\dfrac{E_{a}}{RT}}[/tex]

Here, [tex]e^{-\dfrac{E_{a}}{RT}}[/tex]=fraction of collision

We need to calculate the fraction of collisions

Using formula of fraction of collisions

[tex]f=e^{-\dfrac{E_{a}}{RT}}[/tex]

Where f = fraction of collision

E = activation energy

R = gas constant

T = temperature

Put the value into the formula

[tex]f=e^{-\dfrac{88.20}{8.314\times10^{-3}\times(47+273)}}[/tex]

[tex]f=4.00\times10^{-15}[/tex]

Hence, The fraction of collision is [tex]4.00\times10^{-15}[/tex]

Final answer:

The magnitude of the average velocity equals the average speed when the direction of motion doesn't change and the speed is constant. These conditions are met in straightforward travel without directional change or speed variations. For round trips or trips involving direction changes, the average speed may differ from the magnitude of the average velocity.

Explanation:

The conditions under which the magnitude of the average velocity equals the average speed occur when the motion does not involve a change in direction. The average speed is calculated by dividing the total distance traveled by the elapsed time, while the magnitude of the average velocity is the total displacement divided by the elapsed time. For these two quantities to be the same, the direction of travel must remain constant, meaning there is no reversal or change in direction.

When you take a road trip and do not change direction, and your speed is consistent, then your average speed is equal to the magnitude of the average velocity. However, if you ended up back at your starting point, despite having moved, your displacement would be zero, and hence so would your average velocity, even though your average speed is greater than zero.

If you're calculating the ratio of the total distance as shown on the car's odometer to the time of the trip, you're calculating average speed. The speedometer of a car measures instantaneous speed, not velocity, because it does not provide information about direction.

Answer:

(a). [tex]x=2.00\sin(3.00\pi t)[/tex], hence proved

(b). The maximum speed is 18.8 cm/s.

(c). The maximum  acceleration is 177.65 cm/s².

(d). The total distance is 12 cm.

Explanation:

Given that,

Amplitude = 2.00 cm

Frequency = 1.50 Hz

Given equation of position of the particle is

[tex]x=2.00\ sin(3.00\pit)[/tex]

(a). show that the position of the particle is given by

[tex]x=2.00\sin(3.00\pi t)[/tex]

We know the general equation of S.H.M

[tex]x=A\sin(\omega t[/tex]...(I)

At t =0, x = 0

On differentiating equation (I)

[tex]v=\dfrac{dx}{dt}[/tex]

[tex]v=A\omega\cos(\omega t)[/tex]

At t = 0, the particle moving to the right

[tex]V=A\omega[/tex] > 0

Given statement is true.

The equation of position is

[tex]x=A\sin(\omega t)[/tex]

here, [tex]\Omega= 2\pi f[/tex]

Put the value in the equation

[tex]x=2.00\sin(2\times1.50\pi t)[/tex]

[tex]x=2.00\sin(3.00\pi t)[/tex]

Hence proved.

(b). We need to calculate the maximum speed

[tex]V=A\omega\cos(\omega t)[/tex]....(II)

At t = 0,

[tex]V_{max}=A\omega[/tex]

Put the value into the formula

[tex]V_{max}=2.00\times2\pi\times1.50[/tex]

[tex]V_{max}=6\pi[/tex]

[tex]V_{max}=18.8\ cm/s[/tex]

(c). We need to calculate the maximum  acceleration

Using equation (II)

[tex]V=A\omega\cos(\omega t)[/tex]

On differentiating

[tex]a=\dfrac{dV}{dt}[/tex]

[tex]a=-A\omega^2\sin(\omega t)[/tex]

[tex]a_{max}\ when\ \sin(\omega t)\ is\ -1[/tex]

[tex]a_{max}=-A\omega^2\times-1[/tex]

[tex]a=A\omega^2[/tex]

[tex]a_{max}=2\times(3\pi)^2\approx 177.65 cm/s^2[/tex]

(d). We need to calculate the total distance traveled between t = 0 and t = 1.00 s

Using equation (II)

[tex]V=A\omega\cos(\omega t)[/tex]

On integration

[tex]\int{V}=\int_{t}^{t'}{A\omega\cos(\omega t)}[/tex]

Put the vale into the formula

[tex]\int{V}=\int_{0}^{1}{A\omega\cos(\omega t)}[/tex]

[tex]D=\int_{0}^{1}|6\pi\cos\left(3\pi t\right)|dt[/tex]

[tex]D=12\ cm[/tex]

Hence, (b). The maximum speed is 18.8 cm/s.

(c). The maximum  acceleration is 177.65 cm/s².

(d). The total distance is 12 cm.

A particle in simple harmonic motion can be described by the equation x = (2.00 cm)sin(3.00πt). The maximum speed and maximum acceleration occur when the particle is at its maximum displacement from the equilibrium position. The total distance traveled between t = 0 and t = 1.00 s is 4.00 cm.

The position of the particle in simple harmonic motion is given by the equation x = (2.00 cm)sin(3.00πt), where x represents the displacement from the equilibrium position and t represents time in seconds.

(b) The maximum speed of the particle is equal to the amplitude of the motion multiplied by the angular frequency, vmax = (2.00 cm)(2π)(1.50 Hz). The earliest time at which the particle reaches this maximum speed is when it passes through its equilibrium position, t = 0.

(c) The maximum acceleration of the particle is equal to the amplitude of the motion multiplied by the angular frequency squared, amax = (2.00 cm)(2π)(1.50 Hz)2. The earliest time at which the particle reaches this maximum acceleration is when it is at its maximum displacement from the equilibrium position, which occurs at t = 0.

(d) To find the total distance traveled between t = 0 and t = 1.00 s, we calculate the area under the velocity versus time graph. Since the particle is in simple harmonic motion, the velocity varies sinusoidally, and the total distance traveled is equal to two times the amplitude of the motion, dtotal = 2(2.00 cm) = 4.00 cm.

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Answer:

11.48 degree N of W

Explanation:

We are given that

Wind velocity=[tex]v_w=-43[/tex]km/h

Because wind is blowing towards south

Air speed=[tex]v_a=-212km/h[/tex]

Because the captain want to move with air speed in west direction.

x component of relative velocity=-212 km/h

y-Component of relative velocity=-43km/h

Direction=[tex]\theta=tan^{-1}(\frac{y}{x})[/tex]

[tex]\theta=tan^{-1}(\frac{-43}{-212}=11.48^{\circ}[/tex]N of W

Hence, the direction in which the pilot should set her course to travel due west=11.48 degree N of W

Answer:

Explanation:

Under constant acceleration the average velocity of a particle is half the sum of its initial and final velocities. Is this still true if the acceleration is not constant? Explain.

Answer:

The magnitude of electric field at the center of each ring is 129.96 N/C

Explanation:

As per the question:

The diameter of the ring , d = 10 cm = 0.1 m

Radius, [tex]r = \frac{d}{2} = \frac{0.1}{2} = 0.05\ m[/tex]

Separation between the rings, d = 20.0 cm = 0.20 m

Charge on a ring, q = +20 nC = [tex]20\times 10^{- 9}\ C[/tex]

Now,

The electric field at the center of either ring is given by:

[tex]E = \frac{1}{4\pi \epsilon_{o}}\frac{qd}{(d^{2} + r^{2})^{\frac{3}{2}}}[/tex]

where

[tex]\frac{1}{4\pi \epsilon_{o}} = 9\times 10^{9}[/tex]

Thus

[tex]E = 9\times 10^{9}\times \frac{20\times 10^{- 9}\times 0.20}{(0.20^{2} + 0.05^{2})^{\frac{3}{2}}}[/tex]

E = 129.96 N/C

Answer:

[tex]W = -2.76\times 10^{-9}~J[/tex]

Explanation:

The work done on the capacitor is equal to the difference in potential energy stored in the capacitor in two different cases.

The potential energy is given by the following formula:

[tex]U = \frac{1}{2}CV^2[/tex]

where C can be calculated using the plate separation and area.

[tex]C = \epsilon\frac{A}{d} = \epsilon\frac{0.0314}{0.01} = 3.14\epsilon[/tex]

Therefore, the potential energy in the first case is

[tex]U = \frac{1}{2}3.14\epsilon (20)^2 = 628\epsilon[/tex]

In the second case:

[tex]C_2 = \epsilon\frac{A}{d} = \epsilon\frac{0.0314}{0.02} = 1.57\epsilon\\U = \frac{1}{2}C_2 V^2 = \frac{1}{2}1.57\epsilon (20)^2 = 314\epsilon[/tex]

The permittivity of the air is very close to that of vacuum, which is 8.8 x 10^-12.

So, the difference in the potential energy is

[tex]W = U_2 - U_1 = \epsilon(314 - 628) = -314 \times 8.8 \times 10^{-12} = -2.76\times 10^{-9}~J[/tex]

Answer:

The electron began in the quantum level of 4

Explanation:

Using the formula of wave number:

Wave Number = 1/λ = Rh(1/n1² - 1/n2²)

where,

Rh = Rhydberg's Constant = 1.09677 x 10^7 /m

λ = wavelength of light emitted = 486.4 nm = 486.4 x 10^-9 m

n1 = final shell = 2

n2 = initial shell = ?

Therefore,

1/486.4 x 10^-9 m = (1.09677 x 10^7 /m)(1/2² - 1/n2²)

1/4 - 1/(486.4 x 10^-9 m)(1.09677 x 10^7 /m) = 1/n2²

1/n2² = 0.06082

n2² = 16.44

n2 = 4

The principal quantum level in which the electron began is 5.

Given the following data:

Rydberg constant = [tex]1.09 \times 10^7\; m^{-1}[/tex]

To determine the principal quantum level in which the electron began, we would use the Rydberg equation:

Mathematically, the Rydberg equation is given by the formula:

[tex]\frac{1}{\lambda} = R(\frac{1}{n_f^2} -\frac{1}{n_i^2})[/tex]

Where:

Substituting the parameters into the formula, we have;

[tex]\frac{1}{486.4 \times 10^{-9}} = 1.09 \times 10^7(\frac{1}{2^2} -\frac{1}{n_i^2})\\\\\frac{1}{486.4 \times 10^{-9} \times 1.09 \times 10^7}=\frac{1}{4} -\frac{1}{n_i^2}\\\\\frac{1}{5.3018} =\frac{1}{4} -\frac{1}{n_i^2}\\\\\frac{1}{n_i^2}=\frac{1}{4}-\frac{1}{5.3018}\\\\\frac{1}{n_i^2}=0.25-0.1886\\\\\frac{1}{n_i^2}=0.0614\\\\n_i^2=\frac{1}{0.0614} \\\\n_i=\sqrt{16.29} \\\\n_i=4.0[/tex]

Initial transition = 4.0

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Answer:

(a). The velocity of bus at 2.0 sec is 6.8 m/s.

(b). The position of bus at 2.0 s is 11.8 m.

(c).  [tex]a_{y}-t[/tex], [tex]v_{y}-t[/tex] and x-t graphs

Explanation:

Given that,

[tex]\alha=1.2\ m/s^3[/tex]

Time t = 1.0 s

Velocity = 5.0

The Acceleration equation is

[tex]a_{x(t)}=\alpha t[/tex]

We need to calculate the velocity

Using formula of acceleration

[tex]a=\dfrac{dv}{dt}[/tex]

On integrating

[tex]\int_{v_{0}}^{v}{dv}=\int_{0}^{t}{a dt}[/tex]

Put the value into the formula

[tex]v-v_{0}=1.2\int_{0}^{t}{t dt}[/tex]

[tex]v-v_{0}=0.6t^2[/tex]

[tex]v=v_{0}+0.6t^2[/tex]

Put the value into the formula

[tex]v_{0}=5.0-0.6\times(1.0)^2[/tex]

[tex]v_{0}=4.4\ m/s[/tex]

We need to calculate the velocity at 2.0 sec

Put the value of initial velocity in the equation

[tex]v=4.4+0.6\times(2.0)^2[/tex]

[tex]v=6.8\ m/s[/tex]

(b). If the bus’s position at time t = 1.0 s is 6.0 m,

We need to calculate the position

Using formula of velocity

[tex]v=\dfrac{dx}{dt}[/tex]

On integrating

[tex]\int_{x_{0}}^{x}{dx}=\int_{0}^{t}{v dt}[/tex]

[tex]x_{0}-x=\int_{0}^{t}{v_{0}dt}+\int_{0}^{t}{0.6 t^2}[/tex]

[tex]x_{0}-x=v_{0}t+\dfrac{0.6}{3}t^3[/tex]

[tex]x=x_{0}+v_{0}t+\dfrac{0.6}{3}t^3[/tex]

[tex]x_{0}=6-4.4\times1-\dfrac{0.6}{3}\times1^3[/tex]

[tex]x=1.4\ m[/tex]

The position at t = 2.0 s

[tex]x=1.4+4.4\times2.0+\dfrac{0.6}{3}\times2^3[/tex]

[tex]x=11.8\ m[/tex]

Hence, (a). The velocity of bus at 2.0 sec is 6.8 m/s.

(b). The position of bus at 2.0 s is 11.8 m.

(c).  [tex]a_{y}-t[/tex], [tex]v_{y}-t[/tex] and x-t graphs

Answer:

W = 17,700 lb

Explanation:

given,

Weight limit of the Bridge = 50,000 lb

Weight of empty truck = 19800 lb

Weight on the back of the truck = 12500 lb

Now,

Total weight of truck + cargo

 = Weight of empty truck + Weight on the back of the truck

 = 19800 + 12500

 = 32300 lb

Weight of cargo which is still allowed.

W = weight limit - weight of the system at present

W = 50000 - 32300

W = 17,700 lb

Weight Truck can still carry is equal to W = 17,700 lb

Answer:

Both choices B and C are valid

Explanation:

Sound wave are Mechanical wave. Air (or viscus fluid) is the medium of propagation. Sound is produced by the back and forth vibration of the object. Consider the vibration of object is from left to right then this back and forth vibrations of object displaces the molecules of the medium both rightward and leftward (to and fro) to the direction of the energy transport forming compression and rarefaction. This shows that the motion of the molecules of the medium is both parallel (and anti-parallel) to the direction of the sound wave propagation.

Final answer:

The motion of the particles of a medium in a sound wave is parallel to the direction of the wave motion.

Explanation:

A sound wave is a longitudinal wave, which means that the motion of the particles of the medium is parallel to the direction of the wave motion. In other words, the particles of the medium vibrate or oscillate back and forth in the same direction that the sound wave is traveling. This can be compared to compressing and stretching a coiled spring. As the wave propagates through the medium, it creates zones of compression and rarefaction, causing the air molecules to move in the same direction as the sound wave.

Answer:

[tex]E' = \frac{1}{8} E[/tex]

Explanation:

Given data:

first case

Distance of electric field from center of sphere is r_1 <R

Electric field at r_1< R

[tex]E = \frac{kQr_1}{R^3}[/tex]

second case

Distance of electric field from centre of sphere is r_1 < 2R

Electric field at r_1< 2R

[tex]E' = \frac{kQr_1}{8R^3}[/tex]

so, we have

[tex]E' = \frac{1}{8} E[/tex]

The volumetric rate or flow rate of a fluid is defined as the amount of the volume of a fluid circulating on a surface per unit of time. In this case we have units given initially: Gallons and minutes. For the first part we will convert the minutes to seconds, and we will obtain the flow rate under that measure. For the second case we will convert the gallons to cubic meters and obtain the desired value. Recall the following conversion rates,

[tex]1 min = 60s[/tex]

[tex]1 U.S Gal = 0.00378541178 m^3[/tex]

If the flow rate is defined as the volume by time, the flow rate with the given values is

[tex]Q = \frac{V}{t}[/tex]

[tex]Q = \frac{50Gal}{8min}[/tex]

[tex]Q = 6.25 Gal/min[/tex]

PART A ) Converting to Gal/seconds, we have,

[tex]Q = 6.25 \frac{Gal}{min}(\frac{1min}{60s})[/tex]

[tex]Q = 0.10416Gal/s[/tex]

PART B) Converting Gal/seconds to [tex]m^3/s[/tex]

[tex]Q = 0.104116\frac{Gal}{s} (\frac{0.00378541178 m^3}{1 Gal})[/tex]

[tex]Q = 3.941*10^{-4}m^3/s[/tex]

The rate of filling the gasoline tank is 0.104 gallons per second or approximately 0.000383 cubic meters per second.

To calculate the rate at which the gasoline tank is being filled, we need to first convert the given quantities into the relevant units. Given that the gasoline tank is 50.0 gallons and it takes 8.00 minutes to fill it, the flow rate is 50/480 (since 8 min = 480 s) = 0.104 gallons/second. Since 1 US gallon = 231 cubic inches, this gives us a flow rate of 0.104 x 231 = 23.5 cubic inches per second.

To convert this to the rate in cubic meters per second, we use the fact that 1 inch = 0.254 cm and 1 cubic meter = 1,000,000 cubic cm. Therefore, 23.5 cubic inches = 23.5 x 0.254^3 cubic meters = approximately 0.000383 cubic meters per second (m^3/s).

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Answer:

a)The initial velocity of the puck is 20 ft/s.

b)The coefficient of friction is 0.062.

Explanation:

Hi there!

a)For this problem let's use the equations of position and velocity of an object moving in a straight line with constant acceleration:

x = x0 + v0 · t + 1/2 · a · t²

v = v0 + a · t

Where:

x = position of the puck after a time t.

x0 = initial position.

v0 = initial velocity.

a = acceleration.

t = time.

v = velocity of the puck at a time t.

Let's place the origin of the frame of reference at the point where the puck is hit so that x0 = 0.

We know that at t = 10 s the velocity of the puck is zero (v = 0) and its position is 100 ft (x = 100 ft):

100 ft = v0 · 10 s + 1/2 · a · (10 s)²

0 = v0 + a · 10 s

We have a system of two equations with two unknowns, so, we can solve the system.

Solving for v0 in the second equation:

0 = v0 + a · 10 s

v0 = -a · 10 s

Replacing v0 in the first equation:

100 ft = (-a · 10 s) · 10 s + 1/2 · a · (10 s)²

100 ft = -50 s² · a

100 ft / -50 s² = a

a = -2.0 ft/s²

Then the initial velocity of the puck will be:

v0 = -a · 10 s

v0 = -(-2.0 ft/s²) · 10 s

v0 = 20 ft/s

The initial velocity of the puck is 20 ft/s.

b) The friction force is calculated as follows:

Fr = N · μ

Where:

Fr = friction force.

N = normal force.

μ = coefficient of friction.

Since the only vertical forces acting on the puck are the weight of the puck and the normal force and since the puck is not being accelerated in the vertical direction, then, the normal force is equal to the weight of the puck. The weight (W) is calculated as follows:

W = m · g

Where "m" is the mass of the puck and "g" is the acceleration due to gravity (32.2 ft/s²).

Then the friction force can be calculated as follows:

Fr = m · g · μ

Since the acceleration of the puck is provided only by the friction force, then, due to Newton's second law:

Fr = m · a

Where "m" is the mass of the puck and "a" its acceleration. Then:

Fr = m · g · μ

Fr = m · a

m · g · μ = m · a

μ = a/g

μ = 2.0 ft/s² / 32.2 ft/s²

μ = 0.062

The coefficient of friction is 0.062.

Answer:

[tex]F=3.2345*10^{-10}N[/tex]

Explanation:

Given data

Distance r=7.5×10⁻⁹m

Charge of electron -e= -1.6×10⁻¹⁹C

Charge of proton e=1.6×10⁻¹⁹C

To find

Electric force F

Solution

From Coulombs law we know that:

[tex]F=K\frac{q_{1}q_{2} }{r^{2} }[/tex]

q₁ is charge of electron

q₂ is the charge of gold nucleus which contains 79 positively charge protons and 118 neutral neutrons.  

The Charge of single proton e=1.6×10⁻¹⁹C

79 proton charge q₂=79×1.6×10⁻¹⁹=1.264×10⁻¹⁷C

So

[tex]F=\frac{1}{4\pi *8.85*10^{-12} } \frac{-1.6*10^{-19}*1.264*10^{-17}}{(7.5*10^{-9})^{2} }\\ F=3.2345*10^{-10}N[/tex]

The magnitude of the electric force exerted by the gold nucleus on the electron [tex]3.235\times10^{-10}\rm N[/tex].

Electric force is the force of attraction or repulsion between two charged particles. It can be given as,

[tex]F=K\dfrac{q_1\times q_2}{r^2}[/tex]

Here [tex]k[/tex] is coulomb's constant, [tex]q[/tex] is charge on the objects and [tex]r[/tex]  is the distance between two objects.

Given information-

The number of proton in gold atom is 79.

The number of neutrons in gold atom is 118.

The distance of the electron from the nucleus is [tex]7.5 \times 10^{-9} \rm m[/tex].[tex]r[/tex]

The charge on electron is [tex]-1.6\times 10^{-19} C[/tex] and the charge on the proton is [tex]1.6\times 10^{-19} C[/tex].

Put the values in the above equation as,

[tex]F=8.98\times10^9\times\dfrac{(79\times1.6\times10^{-19})\times1.6\times10^{-19}}{(7.5\times10^{-19})^2}\\F=3.235\times10^{-10}\rm N[/tex]

Hence the magnitude of the electric force exerted by the gold nucleus on the electron [tex]3.235\times10^{-10}\rm N[/tex].

Learn more about the electric force here;

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Answer:

Electric field, [tex]E=6.02\times 10^{-3}\ N/C[/tex] to the west direction.                  

Explanation:

Given that,

Acceleration of the electron, [tex]a=1.06\times 10^9\ m/s^2[/tex] (eastwards)

We need to find the magnitude and direction of the electric field. From Newton's law and electrostatic force,

ma = qE

[tex]E=\dfrac{ma}{q}[/tex]

[tex]E=\dfrac{9.1\times 10^{-31}\times 1.06\times 10^9}{1.6\times 10^{-19}}[/tex]

[tex]E=6.02\times 10^{-3}\ N/C[/tex]

The direction of electric field is in opposite direction of the acceleration of the electron. So, the electric field is acting in west direction.

The magnitude of the electric field accelerating an electron eastward is 6.04 × 10⁴ N/C, and its direction is westward since the electron has a negative charge.

To determine the magnitude and direction of the electric field that accelerates an electron eastward, we can use the formula F = qE, where F is the force, q is the charge of the electron, and E is the electric field strength. The force exerted on the electron can be found using Newton's second law, F = ma, where m is the mass of the electron (9.11 × 10⁻³¹ kg) and a is the acceleration (1.06 × 10⁹ m/s²). Thus, F = (9.11 × 10⁻³¹ kg)(1.06 × 10⁹ m/s²) = 9.66 × 10⁻¹¹ N. We then solve for E by rearranging the formula to E = F/q, with q being the charge of an electron (-1.60 × 10⁻¹¹ C). Thus, E = (9.66 × 10⁻¹¹ N) / (-1.60 × 10⁻¹¹ C) = -6.04 × 10⁴ N/C. The negative sign indicates that the electric field points westward since it accelerates the electron eastward, and the electron has a negative charge.

Answer:

0.00461031264 m/s

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

M = Mass of the Earth =  6 × 10²⁴ kg

r = Distance between Earth and Sun = [tex]1.5\times 10^{11}\ m[/tex]

t = Time taken = 3 days

Acceleration is given by

[tex]a=\dfrac{GM}{r^2}\\\Rightarrow a=\dfrac{6.67\times 10^{-11}\times 6\times 10^{24}}{(1.5\times 10^{11})^2}\\\Rightarrow a=1.77867\times 10^{-8}\ m/s^2[/tex]

Velocity of the star

[tex]v=u+at\\\Rightarrow v=0+1.77867\times 10^{-8}\times 3\times 24\times 60\times 60\\\Rightarrow v=0.00461031264\ m/s[/tex]

The Sun's speed will be 0.00461031264 m/s

Answer:

[tex] x_2 = 648.46\ m[/tex]

Explanation:

given,

Average velocity due west =  1.20 m/s

case 1

Distance moved in west, x₁ = 5.63 km

speed due west, v₁ = 2.33 m/s

Case 2

Distance moved in east = x₂

speed due east, v₂ = 0.374 m/s

total distance = x₁ + x₂ = 5.62

total time = t₁ + t₂ = 5630/2.33 + x₂/0.374

now,

[tex]average\ velocity = \dfrac{total\ distance}{total\ time}[/tex]

[tex]-1.20= \dfrac{-5630 + x_2}{\dfrac{5630}{2.33}+\dfrac{x_2}{0.374}}[/tex]

negative sign is used because we want the distance in east but velocity is in west.

[tex] - 2900 - 3.21 x_2 = -5630 + x_2[/tex]

[tex]-4.21 x_2 = -2730[/tex]

[tex] x_2 = 648.46\ m[/tex]

The distance she walked in east is equal to 648.46 m

Answer:

The magnitude of the electric field will decrease

Explanation:

The capacitance of a parallel plate capacitor having plate area A and plate separation d is C=ϵ0A/d.  

Where ϵ0 is the permittivity of free space.  

A capacitor filled with dielectric slab of dielectric constant K, will have a new capacitance C1=ϵ0kA/d

        C1=K(ϵ0A/d)

        C1=KC

Where C is the capacitance with no dielectric.

The new capacitance is k times the capacitance of the capacitor without dielectric slab.  

This implies that the charge storing capacity of a capacitor increases k times that of the capacitor without dielectric slab.

The charge stored in the original capacitor Q=CV

The charge stored in the original capacitor after inserting dielectric  Q1=C1V1

The law of conservation of energy states that the energy stored is constant:

i.e Charge stored in the original capacitor is same as charge stored after the dielectric is inserted.

Q   = Q1

CV = C1V1

  CV = C1V1  -------2

We derived C1=KC. Inserting this into equation 2

   CV = KCV1

 V1 = (CV)/KC

         V/K

This implies the voltage decreases when a dielectric is used within the plate.

The relationship between electric field and potential voltage is a linear one

V= Ed

Therefore the electric field will decrease

Final answer:

The capacitance of a parallel-plate capacitor with non-uniform electric field due to increased plate separation will be less than the ideal scenario calculated by C = € A/d.

Explanation:

When the separation of the plates in a parallel-plate capacitor is not small enough to maintain a uniform electric field, the field lines will bulge outwards at the edges. This bulging implies that some of the field lines that emanate from one plate do not reach the opposite plate, reducing the effective area over which the field is acting. Consequently, this non-uniform field means that the capacitance of the capacitor will be less than the ideal calculation using the formula C = € A/d, where C is the capacitance, € is the permittivity of the medium between the plates, A is the plate area, and d is the separation between the plates.

Answer:

q = 8.85 x 10⁻¹¹ C

Explanation:

given,

Electric field, E = 1.18 N/C

distance, r = 0.822 m

Charge magnitude = ?

using formula of electric field.

[tex]E = \dfrac{kq}{r^2}[/tex]

k is the coulomb constant

[tex]q= \dfrac{Er^2}{k}[/tex]

[tex]q= \dfrac{1.18\times 0.822^2}{9\times 10^9}[/tex]

  q = 8.85 x 10⁻¹¹ C

The magnitude of charge is equal to q = 8.85 x 10⁻¹¹ C

Answer:

∆u =-111.8 kJ/kg

T_fin=-10.09℃

Explanation:

note:

solution is attached in word file due to some technical issue in mathematical equation. please find the attached documents.

Answer:

725.2‬ N

Explanation:

Since it is not stated the scale, the person or both accelerated or experience weightlessness, the net force acting on the bathroom scale is the weight of the person acting downward as the person stands on the scale .

                       Weight = mass of a body × acceleration due to gravity

                                    = 74 kg × 9.8 m/s²

                                    = 725.2‬ N    

Answer:

[tex]V = 1.44\times 10^{5}~V[/tex]

Explanation:

The electric potential can be found by using the following formula

[tex]V = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}[/tex]

Applying this formula to each charge gives the total potential.

[tex]V = V_1 + V_2 + V_3 + V_4\\V = \frac{1}{4\pi\epsilon_0}\frac{4\times 10^{-6}}{(\sqrt{2}/2)^2} + \frac{1}{4\pi\epsilon_0}\frac{4\times 10^{-6}}{(\sqrt{2}/2)^2} + \frac{1}{4\pi\epsilon_0}\frac{3\times 10^{-6}}{(\sqrt{2}/2)^2} - \frac{1}{4\pi\epsilon_0}\frac{3\times 10^{-6}}{(\sqrt{2}/2)^2}\\V = \frac{16\times 10^{-6}}{4\pi\epsilon_0}\\V = 1.44\times 10^{5}~V[/tex]

Since the potential is a scalar quantity, it is safe to sum all the potentials straightforward. And since they all placed on the corners of a square, +3 and -3 μC charges cancel out each other.