A solenoid 10.0 cm in diameter and 70.0 cm long is made from copper wire of diameter 0.100 cm, with very thin insulation. The wire is wound onto a cardboard tube in a single layer, with adjacent turns touching each other. To produce a field of 5.10 mT at the center of the solenoid, what power must be delivered to the solenoid?

Explanation:

We know

the frequency of the tube with one end open and the other end closed follows the given relations as

[tex]f_{1}[/tex] : [tex]f_{2}[/tex] : [tex]f_{3}[/tex] : [tex]f_{4}[/tex] = 1 : 3 : 5 : 7

∴ the 4th allowed wave is [tex]f_{4}[/tex] = 7 [tex]f_{1}[/tex]

                                                                   = [tex]\frac{7v}{4l}[/tex]

We know  [tex]f_{4}[/tex] = 1975 Hz and v = 343 m/s ( as given in question )

∴[tex]l = \frac{7\times v}{4\times f_{4}}[/tex]

 [tex]l = \frac{7\times 343}{4\times 1975}[/tex]

           = 0.303 m

We know that v = [tex]f_{4}[/tex] x [tex]λ_{4}[/tex]

[tex]\lambda _{4}= \frac{v}{f_{4}}[/tex]

[tex]\lambda _{4}= \frac{343}{1975}[/tex]

                            = 0.17 m

Now when the warmer air is flowing, the speed gets doubled and the mean temperature increases. And as a result the wavelength increases but the amplitude and the frequency remains the same.

So we can write

v ∝ λ

or  [tex]\frac{v_{1}}{v_{2}}= \frac{\lambda _{1}}{\lambda _{2}}[/tex]

Therefore, the wavelength becomes doubled = 0.17 x 2

                                                                             = 0.34 m

Now the new length of the air column becomes doubled

∴ [tex]l^{'}[/tex] = 0.3 x 2

                        = 0.6 m

∴ New speed, [tex]v^{'}[/tex] = 2 x 343

                                               = 686 m/s

∴ New frequency is [tex]f^{'}=\frac{v^{'}}{4\times l^{'}}[/tex]

                                 [tex]f^{'}=\frac{686}{4\times 0.6}[/tex]

                                               = 283 Hz

∴ The new frequency remains the same.

Now we know

[tex]v_{s}[/tex] = 12 m/s, [tex]v_{o}[/tex] = 4 m/s, [tex]f_{o}[/tex] = 1975 Hz

Therefore, apparent frequency is [tex]f^{'}=f^{o}\left ( \frac{v+v_{s}}{v+v_{o}} \right )[/tex]

[tex]f^{'}=1975\left ( \frac{343+12}{343+4} \right )[/tex]

            = 2020.5 Hz

Answer:

(a) 250 kg/m^3

(b) 2 m^3

(c) 1000 kg

Explanation:

(a) Let V be the total volume of the boat.

Volume immersed = 25 % of total volume = 25 V / 100   = V / 4

Let the density of boat is d.

Density of water = 1000 kg/m^3

Use the principle of flotation

Buoyant force on the boat = weight of the boat

Volume immersed x density of water x g = Volume x density of boat x g

V / 4 x 1000 x g = V x d x g

d = 250 kg/m^3

(b)

mass = 500 kg

density = 250 kg/m^3

V = mass / density = m / d = 500 / 250 = 2 m^3

(c) Let the mass of cargo is m.

Volume immeresed = 75 % of total volume = 75 x 2 / 100 = 1.5 m^3

Buoyant force = weight of boat + weight of cargo

1.5 x 1000 x g = 500 g + m g

1500 = 500 + m

m = 1000 kg

in increasing order of kinetic energy, the joggers are: A, D, C, B

To rank the joggers in order of increasing kinetic energy, we'll use the formula for kinetic energy:

[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]

Where:

- ( m ) is the mass of the jogger

- ( v ) is the speed of the jogger

Let's calculate the kinetic energy for each jogger:

1. Jogger A:

[tex]\[ KE_A = \frac{1}{2} m v^2 \][/tex]

2.Jogger B:

[tex]\[ KE_B = \frac{1}{2} \left(\frac{m}{3}\right) (3v)^2 = \frac{1}{2} \left(\frac{m}{3}\right) 9v^2 = \frac{9}{2} \left(\frac{1}{3} m v^2\right) = \frac{9}{2} KE_A \][/tex]

3. Jogger C:

[tex]\[ KE_C = \frac{1}{2} (4m) \left(\frac{v}{3}\right)^2 = \frac{1}{2} (4m) \left(\frac{1}{9}\right) v^2 = \frac{2}{9} (4m v^2) = \frac{8}{9} KE_A \][/tex]

4. Jogger D:

[tex]\[ KE_D = \frac{1}{2} (3m) \left(\frac{v}{2}\right)^2 = \frac{1}{2} (3m) \left(\frac{1}{4}\right) v^2 = \frac{3}{8} (m v^2) = \frac{3}{4} KE_A \][/tex]

Now, let's rank the joggers based on their kinetic energies:

- Jogger A: [tex]\( KE_A \)[/tex]

- Jogger D: [tex]\( KE_D = \frac{3}{4} KE_A \)[/tex]

- Jogger C: [tex]\( KE_C = \frac{8}{9} KE_A \)[/tex]

- Jogger B: [tex]\( KE_B = \frac{9}{2} KE_A \)[/tex]

So, in increasing order of kinetic energy, the joggers are: A, D, C, B

Answer:

2.15 m

Explanation:

u = 2.2 m/s, v = 0, uk = 0.115

a = uk x g = 0.115 x 9.8 = 1.127 m/s^2

Let s be the distance traveled before stopping.

v^2 = u^2 - 2 a s

0 = 2.2^2 - 2 x 1.127 x s

s = 2.15 m

Answer:

(a) 4323.67 N

(b) 421.13 degree

Explanation:

A = 2580 N 36 degree South of east

B = 2270 N due South

A = 2580 ( Cos 36 i - Sin 36 j) = 2087.26 i - 1516.49 j

B = 2270 (-j) = - 2270 j

A + B = 2087.26 i - 1516.49 j - 2270 j

A + B = 2087.26 i - 3786.49 j

(a) Magnitude of A + B = [tex]\sqrt{2087.26^{2}+(-3786.49)^{2}}[/tex]

Magnitude of A + B = 4323.67 N

(b) Direction of A + B

Tan theta = - 3786.49 / 2087.26

theta = - 61.13 degree

Angle from ast = 360 - 61.13 = 421.13 degree.

To determine the magnitude and direction of the resultant force acting on a tree stump when two forces are applied, one must resolve the forces into components, sum the components, then apply the Pythagorean theorem for magnitude and inverse tangent for direction.

We need to resolve each force into its horizontal (east-west) and vertical (north-south) components. For Force A, we can use trigonometry to find the components:

Force B is already pointing due south, so its components are:

The net force components are:

With these components, the magnitude of the resultant force can be calculated using the Pythagorean theorem:

Fnet = √(Fnet,x² + Fnet,y²)

To find the direction of the resultant force relative to due east, we can use the inverse tangent function (tan-1):

θ = tan-1(Fnet,y / Fnet,x)

This angle should be positive, as we are measuring it with respect to due east.

Answer:

(a) 15.4 second

(b) 769.23 m

Explanation:

u = 0, a = 6.5 m/s^2, v = 100 m/s

(a) Let t be the time taken

Use First equation of motion

v = u + a t

100 = 0 + 6.5 x t

t = 15.4 second

(b) Let height covered is h.

Use third equation of motion

v^2 = u^2 + 2 a h

100^2 = 0 + 2 x 6.5 x h

10000 = 13 x h

h = 769.23 m

Answer:

[tex]P_{1}-P_{2}=5400lb/ft^{2}[/tex]

Explanation:

We shall use newtons second law to evaluate the pressure difference

For the system the forces that act on it as shown in the figure

Thus by Newton's second law

[tex]F_{1}-F_{2}=mass\times acceleration\\\\P_{1}\times Area-P_{2}\times Area=mass\times acceleration\\\\\because Force=Pressure\times Area\\\\\therefore P_{1}-P_{2}=\frac{mass\times acceleration}{Area}[/tex]

Mass of the gasoline can be calculated from it's density[tex]45lb/ft^{3}[/tex]

[tex]Mass=Density\times Volume\\\\Mass= 45lb/ft^{3}\times \pi \frac{d^{2}}{4}L\\\\Mass=45lb/ft^{3}\times\frac{\pi 8^{2}}{4}\times 24\\Mass=54286.72lbs[/tex]

Using the calculated values we get

[tex]P_{1}-P_{2}=\frac{54286.72\times 5}{\frac{\pi 8^{2}}{4}}[/tex]

[tex]P_{1}-P_{2}=5400lb/ft^{2}[/tex]

Answer:

option (b)

Explanation:

According to the Pascal's law

F / A = f / a

Where, F is the force on ram, A be the area of ram, f be the force on plunger and a be the area of plunger.

Diameter of ram, D = 20 cm, R = 20 / 2 = 10 cm

A = π R^2 = π x 100 cm^2

F = 3 tons = 3000 kgf

diameter of plunger, d = 3 cm, r = 1.5 cm

a = π x 2.25 cm^2

Use Pascal's law

3000 / π x 100 = f / π x 2.25

f = 67.5 Kgf

Answer:

670400 J

Explanation:

m = mass of water = 2 kg

T₀ = initial temperature of water = 20 ºC

T = initial temperature of water = 100 ºC

c = heat capacity of water = 4190 J /kg ºC

Q = energy needed to raise the temperature of water

Energy needed to raise the temperature of water is given as

Q = m c (T - T₀)

Inserting the values

Q = (2) (4190) (100 - 20)

Q = 670400 J

(a) 107.2 Ω

The voltage in the circuit is written in the form

[tex]V=V_0 sin(\omega t)[/tex]

where in this case

V0 = 35.0 V is the maximum voltage

[tex]\omega=100 rad/s[/tex]

is the angular frequency

Given the inductance: L = 165 mH = 16.5 H, the reactance of the inductance is:

[tex]X_L = \omega L=(100)(0.165)=16.5 \Omega[/tex]

Given the capacitance: [tex]C=99\mu F=99\cdot 10^{-6} F[/tex], the reactance of the capacitor is:

[tex]X_C = \frac{1}{\omega C}=\frac{1}{(100)(99\cdot 10^{-6})}=101.0 \Omega[/tex]

And given the resistance in the circuit, R = 66.0 Ω, we can now find the impedance of the circuit:

[tex]Z=\sqrt{R^2+(X_L-X_C)^2}=\sqrt{66^2+(16.5-101.0)^2}=107.2\Omega[/tex]

(b) 0.33 A

The maximum current can be calculated by using Ohm's Law for RLC circuit. In fact, we know the maximum voltage:

V0 = 35.0 V

The equivalent of Ohm's law for an RLC circuit is

[tex]I=\frac{V}{Z}[/tex]

where

Z=107.2 Ω

is the impedance. Substituting these values into the formula, we find:

[tex]I=\frac{35.0}{107.2}=0.33 A[/tex]

(c) 100 rad/s

The equation for the current is

[tex]I=I_0 sin(\omega t-\Phi)[/tex]

where

I0 = 0.33 A is the maximum current, which we have calculated previously

[tex]\omega[/tex] is the angular frequency

[tex]\Phi[/tex] is the phase shift

In an RLC circuit, the voltage and the current have the same frequency. Therefore, we can say that

[tex]\omega=100 rad/s[/tex]

also for the current.

(d) -0.907 rad

Here we want to calculate the phase shift [tex]\Phi[/tex], which represents the phase shift of the current with respect to the voltage. This can be calculated by using the equation:

[tex]\Phi = tan^{-1}(\frac{X_L-X_C}{R})[/tex]

Substituting the values that we found in part a), we get

[tex]\Phi = tan^{-1}(\frac{16.5-101.0}{66})=-52^{\circ}[/tex]

And the sign is negative, since the capacitive reactance is larger than the inductive reactance in this case. Converting in radians,

[tex]\theta=-52 \cdot \frac{2\pi}{360}=-0.907 rad[/tex]

So the complete equation of the current would be

[tex]I=0.33sin(100t+0.907)[/tex]

The impedance of the circuit is 107.2 Ω.

The maximum current is 0.33 A.

The numerical value for ω in the equation i = Imax sin (ωt − ϕ). rad/s is 100 rad/s.

The numerical value for ϕ in the equation i = Imax sin (ωt − ϕ) is -0.907 rad.

1. To find the voltage, we already know that V0=35V

And the voltage in the circuit is V= V0sin(wt)

Hence, the angular frequency, w= 100 rad/s

Given the inductance is 16.5Ω and the capacitance is 101Ω.

And the resistance is 66Ω

Hence, the impedance of the circuit is [tex]\sqrt{66^{2} + (16.5-101)^2} =107.2[/tex]Ω

2. To find the maximum current, since Z= 107.2Ω,

Recall, I = V/Z

Put the values into the formula

I = 0.33A.

3. To find the numerical value for ω in the equation i = Imax sin (ωt − ϕ)
We recall that I0 = 0.33A

In an RLC circuit, the voltage and the current have the same frequency. Therefore, we can say that

w= 100 rad/s.

4. To find the numerical value for ϕ in the equation i = Imax sin (ωt − ϕ)

Because the sign is in the negative and since the capacitive reactance is larger than the inductive reactance in this case.

Converting in radians,

θ = -52. 2π/360

= -0.907rad.

Hence, the complete equation would be:

I = 0.33sin(100t + 0.907)

Read more about impedance here:

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Answer:

1.1 sec

Explanation:

m = mass of the box = 8 kg

k = spring constant of the spring = 69 N/m

v = initial speed of the box = 1.5 m/s

t = time period of oscillation of box in contact with the spring

Time period is given as

[tex]t = \pi \sqrt{\frac{m}{k}}[/tex]

Inserting the values

[tex]t = (3.14) \sqrt{\frac{8}{69}}[/tex]

t = 1.1 sec

Answer:

Q = 116.8 J

Explanation:

Here given that the temperature of 1 L hydrogen is increased by 90 degree C at constant pressure condition.

So here we will have

[tex]Q = n C_p \Delta T[/tex]

here we know that

n = number of moles

[tex]n = \frac{1}{22.4}[/tex]

[tex]n = 0.0446[/tex]

for ideal diatomic gas molar specific heat capacity at constant pressure is given as

[tex]C_p = \frac{7}{2}R[/tex]

now we have

[tex]Q = (0.0446)(\frac{7}{2}R)(90)[/tex]

[tex]Q = 116.8 J[/tex]

Answer:

The drift velocity of electrons in a copper wire is [tex]1.756\times10^{-4}\ m/s[/tex]

Explanation:

Given that,

Density [tex]\rho=8.93\ g/cm^3[/tex]

Mass [tex]M=63.5\ g/mol[/tex]

Radius = 0.625 mm

Current = 3 A

We need to calculate the drift velocity

Using formula of drift velocity

[tex]v_{d}=\dfrac{J}{ne}[/tex]

Where, n = number of electron

j = current density

We need to calculate the current density

Using formula of current density

[tex] J=\dfrac{I}{\pi r^2}[/tex]

[tex] J=\dfrac{3}{3.14\times(0.625\times10^{-3})^2}[/tex]

[tex] J=2.45\times10^{6}\ A/m^2[/tex]

Now, we calculate the number of electron

Using formula of number of electron

[tex]n=\dfrac{\rho}{M}N_{A}[/tex]

[tex]n=\dfrac{8.93\times10^{6}}{63.5}\times6.2\times10^{23}[/tex]

[tex]n=8.719\times10^{28}\ electron/m^3[/tex]

Now put the value of n and current density into the formula of drift velocity

[tex]v_{d}=\dfrac{2.45\times10^{6}}{8.719\times10^{28}\times1.6\times10^{-19}}[/tex]

[tex]v_{d}=1.756\times10^{-4}\ m/s[/tex]

Hence, The drift velocity of electrons in a copper wire is [tex]1.756\times10^{-4}\ m/s[/tex]

Answer:

12 × 10^12 Nm^2/C

Explanation:

According to the Gauss's theorem

The total electric flux by the enclosed charge is given by

Charge enclosed / €0

So flux by each surface

= Q enclosed / 6 × €0

= 641 / ( 6 × 8.854 × 10^-12)

= 12 × 10^12 Nm^2/C

Answer:

- 0.23

0.56

Explanation:

[tex]\underset{v_{o}}{\rightarrow}[/tex] = initial velocity of the particle = [tex](5.24 Cos35.2) \hat{i} + (5.24 Sin35.2) \hat{j}[/tex] = [tex](4.28) \hat{i} + (3.02) \hat{j}[/tex]

[tex]\underset{v_{f}}{\rightarrow}[/tex] = final velocity of the particle = [tex](6.25 Cos57.5) \hat{i} + (6.25 Sin57.5) \hat{j}[/tex] = [tex](3.36) \hat{i} + (5.27) \hat{j}[/tex]

t = time interval = 4.00 sec

[tex]\underset{a}{\rightarrow}[/tex] = average acceleration = ?

Using the kinematics equation

[tex]\underset{v_{f}}{\rightarrow}[/tex] = [tex]\underset{v_{o}}{\rightarrow}[/tex] + [tex]\underset{a}{\rightarrow}[/tex] t

[tex](3.36) \hat{i} + (5.27) \hat{j}[/tex] = [tex](4.28) \hat{i} + (3.02) \hat{j}[/tex] + [tex]\underset{a}{\rightarrow}[/tex] (4)

[tex](- 0.92) \hat{i} + (2.25) \hat{j}[/tex] = 4 [tex]\underset{a}{\rightarrow}[/tex]

[tex]\underset{a}{\rightarrow}[/tex] = [tex](- 0.23) \hat{i} + (0.56) \hat{j}[/tex]

hence

Average acceleration along x-direction = - 0.23

Average acceleration along y-direction = 0.56

Answer:

0.31 m

Explanation:

m = mass of the block = 1.5 kg

H = height from which the block is released on ramp = 0.81 m

k = spring constant of the spring = 250 N/m

x = maximum compression of the spring

using conservation of energy

Spring potential energy gained by spring = Potential energy lost by block

(0.5) k x² = mgH

(0.5) (250) x² = (1.5) (9.8) (0.81)

x = 0.31 m

Final answer:

The maximum compression of the spring, when a block of mass 1.5 kg is released from rest at a height of 0.81 m on a frictionless ramp onto a spring with spring constant 250.0 N/m, is approximately 30.7 cm.

Explanation:

To determine the maximum compression of the spring, x, we use the conservation of energy principle. The gravitational potential energy (PE) at the height H will convert to the elastic potential energy (spring PE) in the spring when the block of mass m compresses the spring at the foot of the ramp.

Initially, the block has gravitational potential energy given by PE = mgh, where g = 9.81 m/s2 is the acceleration due to gravity. At the point of maximum compression, all this energy will be stored as elastic potential energy in the spring given by spring PE = ½kx2.

Setting the initial PE equal to the spring PE and solving for x, we get:

mgh = ½kx2

1.5 kg × 9.81 m/s2 × 0.81 m = ½ × 250.0 N/m × x2

x2 = (1.5 kg × 9.81 m/s2 × 0.81 m) / (0.5 × 250.0 N/m)

x2 = (11.7915 kg·m2/s2) / (125.0 N/m)

x2 = 0.094332 m2

x = √0.094332 m2

x ≈ 0.307 m or 30.7 cm

Therefore, the maximum compression of the spring is approximately 30.7 cm.

Answer:

Heat generated in the rod is 8.33 watts.

Explanation:

It is given that,

Length of rod, l = 2 m

Area of cross section, [tex]A=2\ mm\times 2\ mm=4\ mm^2=4\times 10^{-6}\ m^2[/tex]

Resistivity of rod, [tex]\rho=6\times 10^{-8}\ \Omega-m[/tex]

Potential difference, V = 0.5 V

The value of resistance is given by :

[tex]R=\rho\dfrac{l}{A}[/tex]

[tex]R=6\times 10^{-8}\ \Omega-m\times \dfrac{2\ m}{4\times 10^{-6}\ m^2}[/tex]

R = 0.03 ohms

Let H is the rate at which heat is generated in the rod . It is given by :

[tex]\dfrac{H}{t}=I^2R[/tex]

Since, [tex]I=\dfrac{V}{R}[/tex]

[tex]\dfrac{H}{t}=\dfrac{V^2}{R}[/tex]

[tex]\dfrac{H}{t}=\dfrac{(0.5)^2}{0.03}[/tex]

[tex]\dfrac{H}{t}=8.33\ watts[/tex]

So, the at which heat is generated in the rod is 8.33 watts. Hence, this is the required solution.

Answer:

Work done, W = 39.2 J

Explanation:

It is given that,

Mass of the block, m = 2 kg

The block is lifted vertically 2 m by the man i.e the distance covered by the block is, h = 2 m. The man is doing work against the gravity. It is given by :

[tex]W=mgh[/tex]

Where

g is acceleration due to gravity

[tex]W=2\ kg\times 9.8\ m/s^2\times 2\ m[/tex]

W = 39.2 J

So, the work done by the man is 39.2 J. Hence, this is the required solution.

Answer:

[tex]P = 2.94 \times 10^5 Pa[/tex]

Explanation:

Normal force due to four tires is counter balancing the weight of the car

So here we will have

[tex]4F_n = mg[/tex]

[tex]F_n = \frac{mg}{4}[/tex]

[tex]F_n = \frac{1.20 \times 10^3 \times 9.81}{4}[/tex]

[tex]F_n = 2943 N[/tex]

now we know that pressure in each tire is given by

[tex]P = \frac{F}{A}[/tex]

Here we know that

[tex]A = 1.00 \times 10^2 cm^2 = 1.00 \times 10^{-2} m^2[/tex]

[tex]P = \frac{2943}{1.00 \times 10^{-2}}[/tex]

[tex]P = 2.94 \times 10^5 Pa[/tex]

Answer:

P = 294300Pa or 42.67psi by conversion.

Explanation:

Since Four tyres were inflated, we have that area of the four tyres are

4×1×10²cm²

Pressure is given as:

P = f/a but f = mg

P = m×g/a

Therefore,

P = 1.20x10³kg × 9.81m/s² / (4 ×1x10² cm²)

P = 1.20x10³kg×9.81m/s² / (0.04m²)

P = 294300Pa or 42.67psi by conversion.

Answer:

1.33 x 10^26 photons

Explanation:

λ = 12 cm = 0.12 m

Δ T = 5 degree C

m = 11 g = 0.011 kg

c = 4 J/g K = 4000 J / kg K

Let the number of photons are n.

Energy of one photon = h c / λ

Where, h is the Plank's constant and c be the velocity of light.

Energy of n photons, E = n h c / λ

This energy is used to raise the temperature of eye, so

n h c / λ = m c Δ T

n = m c Δ T λ / h c

n = (0.011 x 4000 x 5 x 0.12) / (6.63 x 10^-34 x 3 x 10^8)

n = 1.33 x 10^26

Thus, there are 1.33 x 10^26 photons.

To estimate the number of photons that must be absorbed to raise the temperature of an eye by 5.00 °C, calculate the energy required and convert it into the number of photons using Planck's constant and the frequency of the microwaves.

To estimate the number of photons that must be absorbed to raise the temperature of an eye by 5.00 °C, we need to calculate the amount of energy required and then convert it into the number of photons. Here are the steps:

Therefore, approximately 3.30 × 10^11 photons with a wavelength of 12 cm must be absorbed to raise the temperature of an eye by 5.00 °C.

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Answer:

Explanation:

Force, F = - mg j

r = - 7x i + y j

Torque is defined as the product f force and the perpendicular distance.

It is also defined as the cross product of force vector and the displacement vector.

[tex]\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}[/tex]

[tex]\overrightarrow{\tau }=(- 7 x i + yj)\times (-mgj)[/tex]

[tex]\overrightarrow{\tau  }= 7 m g x k

Here, we observe that the torque is independent of y coordinate.

Answer:

i = 0.2 A

Explanation:

R = resistance of the resistor = 1000 Ω

V = Potential difference across the capacitor = Potential difference across the resistor = 200 V

i = initial current just after the capacitor is connected to resistor

Using ohm's law

[tex]i = \frac{V}{R}[/tex]

Inserting the values

[tex]i = \frac{200}{1000}[/tex]

i = 0.2 A

The initial current flowing through a 1000-Ω resistor when a 20.0-μF capacitor charged to 200 V is connected to it is 200 mA.

To find the initial current, we use Ohm's law, which states I = V / R, where I is the current, V is the voltage, and R is the resistance. At the moment the capacitor is connected to the resistor, the voltage across the resistor is the same as the voltage on the capacitor, so we have I = 200 V / 1000 Ω. The initial current is therefore 0.2 A, or 200 mA.

Answer:

[tex]K.E =0.081eV[/tex]

[tex]\vec{\Omega}=(0.54\hat{i}+0.83\hat{j}+0.039\hat{k}[/tex]

Explanation:

Given:

Velocity vector [tex]\vec{v}=(2132\hat{i}+3300\hat{j}+154\hat{k})m/s[/tex]

the mass of neutron, m = 1.67 × 10⁻²⁷ kg

Now,

the kinetic energy (K.E) is given as:

[tex]K.E =\frac{1}{2}mv^2[/tex]

[tex]\vec{v}^2 = \vec{v}.\vec{v}[/tex]

or

[tex]\vec{v}^2 = (2132\hat{i}+3300\hat{j}+154\hat{k}).(2132\hat{i}+3300\hat{j}+154\hat{k})[/tex]

or

[tex]\vec{v}^2 =15.45\times 10^6 m^2/s^2[/tex]

substituting the values in the K.E equation

[tex]K.E =\frac{1}{2}1.67\times 10^{-27}kg\times 15.45\times 10^6 m^2/s^2[/tex]

or

[tex]K.E =1.2989\times 10^{-20}J[/tex]

also

1J = 6.242 × 10¹⁸ eV

thus,

[tex]K.E =1.2989\times 10^{-20}\times 6.242\times 10^{18}[/tex]

[tex]K.E =0.081eV[/tex]

Now, the direction vector [tex]\vec{\Omega}[/tex]

[tex]\vec{\Omega}=\frac{\vec{v}}{\left | \vec{v} \right |}[/tex]

or

[tex]\vec{\Omega}=\frac{(2132\hat{i}+3300\hat{j}+154\hat{k})}{\left |\sqrt{((2132\hat{i})^2+(3300\hat{j})^2+(154\hat{k}))^2} \right |}[/tex]

or

[tex]\vec{\Omega}=\frac{(2132\hat{i}+3300\hat{j}+154\hat{k})}{3930.65}[/tex]

or

[tex]\vec{\Omega}=0.54\hat{i}+0.83\hat{j}+0.039\hat{k}[/tex]

The vertical height between the climbers can be calculated from the projectile motion's kinematic formula: h = (v₀² sin²θ) / 2g. Here, the first aid kit has reached its trajectory's peak when caught, and its vertical velocity component is zero, making it ideal to implement the equation.

In your question, you want to know the vertical height between the two climbers. This is a question in projectile motion in physics. When an object is thrown with an initial velocity at an angle, it creates a trajectory as it travels. When the climber above catches the first aid kit, it's mentioned that it is moving horizontally, which means its vertical speed is zero at that very moment. Hence the first aid kit has reached the maximum height of its trajectory or the peak where the vertical component of the velocity is zero.

We can calculate the vertical distance or height (h) using a formula from kinematics. The formula is: h = (v₀² sin²θ) / 2g. Where v₀ is the initial velocity, θ is the angle of elevation, and g is the acceleration due to gravity (which is ~9.8 m/s² on Earth). Plugging in the values we get: h = (5.70 m/s)² × sin²(45.7 °) / (2 * 9.8 m/s²). Solve this to get the height.

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The vertical height between the two climbers is approximately 0.8616 meters.

The vertical height between the two climbers is given by the equation for the vertical displacement in projectile motion:

[tex]\[ h = \frac{{v_{0y}^2}}{{2g}} \][/tex]

First, we need to find the initial vertical component of the velocity[tex](\( v_{0y} \))[/tex]. Since the initial velocity[tex](\( v_0 \))[/tex] is given as 5.70 m/s at an angle of 45.7 we can use the sine function to find [tex]\( v_{0y} \):[/tex]

[tex]\[ v_{0y} = v_0 \sin(\theta) \][/tex]

[tex]\[ v_{0y} = 5.70 \, m/s \times \sin(45.7) \][/tex]

Now, we calculate the sine of 45.7 and multiply it by the initial velocity:

[tex]\[ v_{0y} = 5.70 \, m/s \times 0.721 \][/tex]

[tex]\[ v_{0y} = 4.1117 \, m/s \][/tex]

Next, we use the equation for vertical displacement to find the height h :

[tex]\[ h = \frac{{(4.1117 \, m/s)^2}}{{2 \times 9.81 \, m/s^2}} \][/tex]

[tex]\[ h = \frac{{16.9046 \, m^2/s^2}}{{19.62 \, m/s^2}} \][/tex]

[tex]\[ h = 0.8616 \, m \][/tex]

Answer:

0.48

Explanation:

m = mass of disc shaped grindstone = 50 kg

d = diameter of the grindstone = 0.52 m

r = radius of the grindstone = (0.5) d = (0.5) (0.52) = 0.26 m

w₀ = initial angular speed = 850 rev/min = 850 (0.10472) rad/s = 88.97 rad/s

t = time taken to stop = 7.5 sec

w = final angular speed  0 rad/s

Using the equation

w = w₀ + α t

0 = 88.97 + α (7.5)

α = - 11.86 rad/s²

F = normal force on the disc by the ax = 160 N

μ = Coefficient of friction

f = frictional force

frictional force is given as

f = μ F

f = 160 μ

Moment of inertia of grindstone is given as

I = (0.5) m r² = (0.5) (50) (0.26)² = 1.69 kgm²

Torque equation is given as

r f = I |α|

(0.26) (160 μ) = (1.69) (11.86)

μ = 0.48

Answer:

v/U=0.79

Explanation:

Given u=[tex]u=\frac{Uy}{\partial }=\frac{Uy}{cx^{1/2}}[/tex]

Now for the given flow to be possible it should satisfy continuity equation

[tex]\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0[/tex]

Applying values in this equation we have

[tex]\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\\\\\frac{\partial u}{\partial x}=\frac{\partial (\frac{Uy}{cx^{1/2}})}{\partial x}\\\\\frac{\partial u}{\partial x}=\frac{-1}{2}\frac{Uy}{cx^{3/2}}\\\\[/tex]

Thus we have

[tex]\frac{\partial v}{\partial y}=\frac{1}{2}\frac{Uy}{cx^{3/2}}\\\\\therefore \int \partial v=\int \frac{1}{2}\frac{Uy}{cx^{3/2}}\partial x\\\\v=\frac{1}{4}\frac{Uy^{2}}{cx^{3/2}}\\\\v=\frac{1}{4}\frac{uy}{x}[/tex] Hence proved [tex]\because u=\frac{Uy}{cx^{1/2}}[/tex]

For maximum value of v/U put y =[tex]\partial[/tex]

[tex]v=\frac{1}{4}\frac{Uy^{2}}{cx^{3/2}}[/tex]

[tex]v=\frac{1}{4}\frac{Uy^{2}}{cx^{3/2}}\\\\\frac{v}{U}=\frac{\partial ^{2}}{4cx^{3/2}}\\\\[/tex]

Thus solving we get using the given values

v/U=0.79

The y-component of velocity in a boundary layer flow is derived using the principle of continuity for an incompressible fluid. Upon evaluation, the ratio v /U has a maximum of 0.25 at the specified location (x = 0.5 m and δ = 5 mm).

The y-component v of the velocity can be solved using the continuity equation, d/dx (u A) + d/dy (v A) = 0, where A is the cross-sectional area of the pipe. This equation arises from the principle of conservation of mass for an incompressible fluid. In this case, our A is the width of the plate (into the page) times the y-distance from the plate or y*b. When the equation derived for velocity, u = Uy/δ, and the equation for the boundary layer thickness, δ = cx1/2, are plugged into the continuity equation and simplified, we derive the expression v = uy/4x.  Substituting for δ, we get v = Uy/(4δ*sqrt(x)), and at x = 0.5 m and δ = 5 mm, the maximum value of v /U is 1/4 or 0.25.

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Answer:

V=4v

Explanation:

To perform this operation, we will refer to Ohm's Law. This Law relates the terms current, voltage and resistance.

The current intensity that passes through a circuit is directly proportional to the voltage or voltage of the circuit and inversely proportional to the resistance it presents.

[tex]I=\frac{v}{r}[/tex]

Where

I=Current

V= Voltage

R=Resistance.

So, in this case:

R=200 ohm

I= 2mA  = 0.002 A

V= searched variable

[tex]V=r*I[/tex]

[tex]V=(200ohm)(0.002A)=4V[/tex]

Explanation:

It is given that,

A mass oscillates up and down on a vertical spring with an amplitude of 3 cm and a period of 2 s. It is a case of simple harmonic motion. If the amplitude of a wave is T seconds, then the distance cover by that object is 4 times the amplitude.

In 2 seconds, distance covered by the mass is 12 cm.

In 1 seconds, distance covered by the mass is 6 cm

So, in 16 seconds, distance covered by the mass is 96 cm

So, the distance covered by the mass in 16 seconds is 96 cm. Hence, this is the required solution.

Answer:

The International Space Station move at 7.22 km/s.

Explanation:

Orbital speed of satellite is given by  [tex]v=\sqrt{\frac{GM}{r}}[/tex], where G is gravitational constant, M is mass of Earth and r is the distance to satellite from centre of Earth.

r = R + h = 6350 + 1400 = 7750 km = 7.75 x 10⁶ m

G = 6.673 x 10⁻¹¹ Nm²/kg²

M = 5.98 x 10²⁴ kg

Substituting

              [tex]v=\sqrt{\frac{6.673\times 10^{-11}\times 5.98\times 10^{24}}{7.75\times 10^6}}=7223.86m/s=7.22km/s[/tex]

  The International Space Station move at 7.22 km/s.      

D = 497.4x10⁻⁶m. The diameter of a mile of 24-gauge copper wire with resistance of 0.14 kΩ and resistivity of copper 1.7×10−8Ω⋅m is 497.4x10⁻⁶m.

In order to solve this problem we have to use the equation that relates resistance and resistivity:

R = ρL/A

Where ρ is the resistivity of the matter, the length of the wire, and A the area of ​​the cross section of the wire.

If a mile of 24-gauge copper wire has a resistance of 0.14 kΩ and the resistivity of copper is 1.7×10⁻⁸ Ω⋅m. Determine the diameter of the wire.

First, we have to clear A from the equation R = ρL/A:

A = ρL/R

Substituting the values

A = [(1.7×10⁻⁸Ω⋅m)(1.6x10³m)]/(0.14x10³Ω)

A = 1.9x10⁻⁷m²

The area of a circle is given by A = πr² = π(D/2)² = πD²/4, to calculate the diameter D we have to clear D from the equation:

D = √4A/π

Substituting the value of A:

D = √4(1.9x10⁻⁷m²)/π

D = 497.4x10⁻⁶m

The diameter of the wire is approximately 4.5 × 10^-6 meters.

To find the diameter of the wire, we can use the formula for resistance:

R = (ρ * L) / (A),

where R is the resistance, ρ is the resistivity, L is the length of the wire, and A is the cross-sectional area of the wire. Rearranging the formula to solve for the area:

A = (ρ * L) / R.

Given that the mile of 24-gauge copper wire has a resistance of 0.14 kΩ (or 0.14 * 10^3 Ω), the resistivity of copper is 1.7 × 10^-8 Ω ⋅ m, and 1 mile is equal to 1.6 km, we can substitute the values in the formula:

A = (1.7 × 10^-8 Ω ⋅ m * (1.6 * 10^3 m)) / (0.14 * 10^3 Ω).

Calculating the area, we find:

A ≈ 1.94 × 10^-11 m^2.

Next, we use the formula for the area of a circle (A = π * r^2) to find the radius (r) of the wire:

r = √(A / π).

Substituting the values, we have:

r ≈ √(1.94 × 10^-11 m^2 / π).

Calculating the radius, we find:

r ≈ 2.25 × 10^-6 m.

Finally, we can double the radius to find the diameter of the wire:

D = 2r ≈ 2 * 2.25 × 10^-6 m = 4.5 × 10^-6 m.

Therefore, the diameter of the wire is approximately 4.5 × 10^-6 meters.

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