In otherwise empty space is a system of 4 particles, each of the same mass. The accelerations of the particles are as follows: a1 = ˆi a a2 = ˆj a 2 a3 = −( ˆi +ˆj) (2 a) a4 = (ˆi −ˆj) (3 a), where a > 0 is a constant. What is the acceleration of the center of mass of the system of 4 particles?

We can determine the angular velocity using the formula ω = v/r, where v is the tangential speed (2.2 m/s) and r is the radius of the circle (510 mm). After converting the radius to meters, we can then plug these values into the formula and solve for ω, yielding an angular velocity of  3.9636 rad/s.

To determine the resulting angular velocity ωOA of rod OA when point B has a displacement s = 220 mm, we can use the following formula:

ωOA = v / r

Where:

ωOA = Angular velocity of rod OA

v = Linear velocity of point B

r = Radius or distance from point O to point B

Given:

v = 2.2 m/s

s = 220 mm = 0.22 m

d = 510 mm = 0.51 m

First, we calculate the radius (r) using the Pythagorean theorem because the motion is along a right-angled triangle:

r² = d² + s²

r² = (0.51 m)² + (0.22 m)²

r² = 0.2601 m² + 0.0484 m²

r² = 0.3085 m²

Now, we calculate the square root of r² to get r:

r = √0.3085 m

r ≈ 0.555 m

Now, we can calculate the angular velocity ωOA:

ωOA = v / r

ωOA = 2.2 m/s / 0.555 m

ωOA ≈ 3.9636 rad/s

The angular velocity ωOA is approximately 3.9636 rad/s. Since the motion is counterclockwise (as indicated by the positive linear velocity of point B), the angular velocity is also positive.

Learn more about Angular Velocity here:

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The energy  stored in the spring when it is compressed 20.0 cm is 15.54J, the maximum height of the ball is 1.056 meters. while the muzzle velocity of the ball is  4.55 m/s.

a) Potential Energy in the Spring

Potential energy = 1/2 * spring constant * (compression distance)^2

Potential energy = 1/2 * 777 N/m * (0.2 m)^2 = 15.54 J

b)  Maximum Height of the Ball

Maximum height = potential energy / weight of the ball

Maximum height = 15.54 J / (1.5 kg * 9.81 m/s^2) = 1.056 m

Therefore, the maximum height reached by the ball is 1.056 meters.

(c) Muzzle Velocity of the Ball

muzzle velocity = sqrt(2 * potential energy / mass)

muzzle velocity = sqrt(2 * 15.54 J / 1.5 kg) = 4.55 m/s

Hence, the muzzle velocity of the ball is 4.55 m/s.

A) The energy stored in the spring when compressed is 15.54 J. B) The maximum height reached by the ball is 1.05 m. C) The muzzle velocity of the ball is 4.56 m/s.

A) The energy stored in a spring, also known as elastic potential energy, can be calculated using the formula:

Espring = ½ * k * x2

where k is the spring constant, and x is the compression distance. By plugging in the given values (k = 777 N/m and x = 0.20 m), we get:

Espring = ½ * 777 N/m * (0.20 m)2 ≈ 15.54 J

B) At the maximum height, all the elastic potential energy is converted into gravitational potential energy (Egravitational), given by:

Egravitational = m * g * h

Where m is the mass of the ball, g is the acceleration due to gravity (9.81 m/s2), and h is the height. Since Espring = Egravitational, we can solve for h:

h = Espring / (m * g) = 15.54 J / (1.5 kg * 9.81 m/s2) ≈ 1.05 m

C) The muzzle velocity of the ball can be found using the conservation of energy principle. The elastic potential energy stored in the spring is converted entirely into the kinetic energy of the ball at the spring's equilibrium position:

Kinetic energy, Ekinetic = ½ * m * v2

Since Espring = Ekinetic, solving for v gives:

v = √(2 * Espring / m) = √(2 * 15.54 J / 1.5 kg) ≈ 4.56 m/s

Answer:

4557.88 m/s

Explanation:

M = mass of earth = 5.98 x 10²⁴ kg

R = radius of earth = 6.4 x 10⁶ m

r = orbital radius of satellite = 3R = 3 x 6.4 x 10⁶ = 19.2 x 10⁶ m

[tex]v[/tex] = speed of the satellite

speed of the satellite is given as

[tex]v=\sqrt{\frac{GM}{r}}[/tex]

[tex]v=\sqrt{\frac{(6.67\times 10^{-11})(5.98\times 10^{24})}{19.2\times 10^{6}}}[/tex]

[tex]v[/tex] = 4557.88 m/s

The index of refraction can be calculated using the speed of light in vacuum divided by the speed of light in a specific medium. Given the speed of light in a medium is 2.26 x 10^8 m/s, the index of refraction would be calculated as roughly 1.33.

To calculate the index of refraction for a certain medium, you would use the speed of light in vacuum (c) divided by the speed of light in that medium (v). This equation is denoted as n = c/v.

The speed of light in vacuum is approximately 3.00×10^8 m/s. Given that the speed of light in a certain substance is 2.26 x 10^8 m/s, we can substitute these values into the equation to find the material's index of refraction.

So, n = (3.00×10^8 m/s) / (2.26 x 10^8 m/s). Therefore, the index of refraction of that substance is approximately 1.33. This property of a substance tells us how much the path of light is bent, or refracted, when entering the material.

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Answer:

156.3 J

Explanation:

E = Amplitude of the electric field = 25000 V/m

t = time interval = 1 minute = 60 sec

d = diameter of the spot = 2 mm = 2 x 10⁻³ m

Area of the spot is given as

A = (0.25) πd²

A = (0.25) (3.14) (2 x 10⁻³)²

A = 3.14 x 10⁻⁶ m²

Energy delivered to the spot is given as

U = (0.5)ε₀ E² c A t

Inserting the values

U = (0.5) (8.85 x 10⁻¹²) (25000)² (3 x 10⁸) (3.14 x 10⁻⁶) (60)

U = 156.3 J

To find the energy delivered by the laser beam in 1 minute, we need to calculate the power of the laser beam and then multiply it by the duration.

E = Amplitude of the electric field = 25000 V/m

t = time interval = 1 minute = 60 sec

d = diameter of the spot = 2 mm = 2 x 10⁻³ m

Area of the spot is given as

A = (0.25) πd²

A = (0.25) (3.14) (2 x 10⁻³)²

A = 3.14 x 10⁻⁶ m²

Energy delivered to the spot is given as

U = (0.5)ε₀ E² c A t

Inserting the values

U = (0.5) (8.85 x 10⁻¹²) (25000)² (3 x 10⁸) (3.14 x 10⁻⁶) (60)

U = 156.3 J

Answer:

[tex]\mu _j=\dfrac{1}{C_p}\left [T\left(\frac{\partial v}{\partial T}\right)_p-v\right]dp[/tex]

Explanation:

Joule -Thompson effect

 Throttling phenomenon is called Joule -Thompson effect.We know that throttling is a process in which pressure energy will convert in to thermal energy.

Generally in throttling exit pressure is low as compare to inlet pressure but exit temperature maybe more or less or maybe remains constant depending upon flow or fluid flow through passes.

Now lets take Steady flow process  

Let

 [tex]P_1,T_1 [/tex] Pressure and temperature at inlet and

 [tex]P_2,T_2 [/tex] Pressure and temperature at exit

We know that Joule -Thompson coefficient given as

[tex]\mu _j=\left(\frac{\partial T}{\partial p}\right)_h[/tex]

Now from T-ds equation

dh=Tds=vdp

So

[tex]Tds=C_pdt-\left [T\left(\frac{\partial v}{\partial T}\right)_p\right]dp[/tex]

⇒[tex]dh=C_pdt-\left [T\left(\frac{\partial v}{\partial T}\right)_p-v\right]dp[/tex]

So Joule -Thompson coefficient

[tex]\mu _j=\dfrac{1}{C_p}\left [T\left(\frac{\partial v}{\partial T}\right)_p-v\right]dp[/tex]

This is Joule -Thompson coefficient for all gas (real or ideal gas)

We know that for Ideal gas Pv=mRT

[tex]\dfrac{\partial v}{\partial T}=\dfrac{v}{T}[/tex]

So by putting the values in

[tex]\mu _j=\dfrac{1}{C_p}\left [T\left(\frac{\partial v}{\partial T}\right)_p-v\right]dp[/tex]

[tex]\mu _j=0[/tex] For ideal gas.

Answer:

220.8 m

30.5 m/s

Explanation:

a = acceleration of the car = 2.1 m/s²

t = time of travel = 14.5 s

x = displacement of the car

v₀ = initial velocity of the car = 0 m/s

Displacement of the car is given as

x = v₀ t + (0.5) a t²

Inserting the values

x = (0) (14.5) + (0.5) (2.1) (14.5)²

x = 220.8 m

v = final velocity of the car

Final velocity of the car is given as

v = v₀ + at

v = 0 + (2.1) (14.5)

v = 30.5 m/s

Final answer:

The magnetic field at a point on the +x-axis near a wire carrying current in the +y-direction will be directed into the page, as determined by the right-hand rule. To cancel the magnetic field at a specific point on the +x-axis, a second wire carrying equal but opposite current must be arranged parallel to the first.

Explanation:

The question concerns the determination of the magnetic field direction around a long, straight wire carrying a current. According to the right-hand rule, if you point your right thumb in the direction of the current, then the direction in which your fingers curl will show the direction of the magnetic field. In this case, the current is running in the +y direction, so if one were to stand at a point on the +x-axis near the wire and apply the right-hand rule, the fingers would curl into the page, indicating that the magnetic field at that point is directed into the page.

An experimenter wanting to make the total magnetic field at the coordinate x = 1.0 m zero must arrange a second wire parallel to the first with current in the opposite direction. This second wire would also need to be carrying the same magnitude of current but in the -y direction to cancel out the magnetic field from the first wire at the point of interest.

To answer this question, let us break down the initial velocity into its horizontal and vertical components.

v_{ix} = vcos(θ)

v_{iy} = vsin(θ)

v_{x} is the horizontal component of the initial velocity, and this value will stay constant over time because we assume only gravity acts on the stone, therefore only the vertical component of the stone's velocity will change over time

v_{iy} is the vertical component of the initial velocity, and this will change over time due to gravity.

v is the magnitude of the initial velocity.

θ is the angle the velocity vector is oriented at with respect to the horizontal.

Given values:

v = 19.6m/s

θ = 40.8°

Plug in these values and solve for v_{ix} and v_{iy}:

v_{ix} = 19.6cos(40.8°) = 14.8m/s

v_{iy} = 19.6sin(40.8°) = 12.8m/s

To find both the horizontal and vertical displacement at any time, we will use this kinematics equation:

D = X + Vt + 0.5At²

When solving for the horizontal displacement, the following values are:

t = elapsed time

D = horizontal displacement

X = initial horizontal displacement

V = initial horizontal velocity

A = horizontal acceleration

There is no initial horizontal displacement, so X = 0m

The initial horizontal velocity V = v_{ix} = 14.8m/s

Assuming we don't care about air resistance, no force has a component acting horizontally on the stone, so A = 0m/s²

Therefore the equation for the stone's horizontal displacement is given by:

D = 14.8t

When solving for the vertical displacement, the following values are:

t = elapsed time

D = vertical displacement

X = initial vertical displacement

V = initial vertical velocity

A = vertical acceleration

There is no initial vertical displacement, so X = 0m

The initial vertical velocity V = v_{iy} = 12.8m/s

Gravity acts downward on the stone, therefore A = -9.81m/s²

Therefore the equation for the stone's vertical displacement is given by:

D = 12.8t - 4.905t²

Now we just plug in various values of t...

a) At t = 1.03s, the horizontal displacement is D = 14.8(1.03) = 15.2m

b) At t = 1.03s, the vertical displacement is D = 12.8(1.03)-4.905(1.03)² = 7.98m

c) At t = 1.73s, the horizontal displacement is D = 14.8(1.73) = 25.6m

d) At t = 1.73s, the vertical displacement is D = 12.8(1.73)-4.905(1.73)² = 7.46m

Before you write down the following results, read the following explanation.

e) At t = 5.05s, the horizontal displacement is D = 14.8(5.05) = 74.7m

f) At t = 5.05s, the vertical displacement is D = 12.8(5.05)-4.905(5.05)² = -60.4m

Notice that we have a negative value for the vertical displacement. This isn't possible within the context of the problem, so the vertical displacement at t = 5.05s is actually 0m

The problem hasn't stated whether the ground is frictionless or not. I'm going to assume it is frictionless and therefore the stone will keep moving across the ground after landing, so the horizontal displacement at t = 5.05s is 74.7m

Answer:

period of rotation is 9.9843 years

Explanation:

Given data

sides  = 1.0×10^12 m

to find out

period of rotation

solution

first we use the gravititional formula i.e

F(g) = 2 F cos 30

here F = G  × mass(sun)² / radius²

F = 6.67 × [tex]10^{-11}[/tex] × (1.99× [tex]10^{30}[/tex])² / (1× [tex]10^{30}[/tex])²

so

F(g) = 2  ×  6.67 × [tex]10^{-11}[/tex] × (1.99× [tex]10^{30}[/tex])² / (1× [tex]10^{30}[/tex])²  ×  (1.73/2)

F(g) = 4.57 ×  [tex]10^{26}[/tex] N

and

by the law of sines

r/sin30 = s/sin120

so r will be

r = 1.0×10^12  (0.5/.87)

r = 5.75 ×  [tex]10^{11}[/tex] m

we know that gravititional force = centripetal force

so  here

centripetal force = mv² /r

centripetal force = 1.99× [tex]10^{30}[/tex] v² / 5.75 ×  [tex]10^{11}[/tex]

so

4.57 ×  [tex]10^{26}[/tex]  = mv² /r

4.57 ×  [tex]10^{26}[/tex]  = 1.99× [tex]10^{30}[/tex] v² / 5.75 ×  [tex]10^{11}[/tex]

v² = 4.57 ×  [tex]10^{26}[/tex] × 5.75 ×  [tex]10^{11} / 1.99× [tex]10^{30}[/tex]

and

v = 11480 m/sec

and we know period of rotation formula

t = d /v

t = 2 [tex]\pi[/tex] r /v

t = 2 × [tex]\pi[/tex] ×  5.75 ×  [tex]10^{11} / 11480

t = 314 × [tex]10^{5} / 31536000 year

t = 9.9843 year

so period of rotation is 9.9843 years

The period of rotation for the three stars can be calculated using Kepler's Third Law and recognizing that the stars form a circular orbital pattern. The relevant distance is from the center of the mass distribution (center of the equilateral triangle) to any of the stars and all three stars have identical masses.

This question relates to astrophysics and can be solved using Kepler's Third Law that provides a relationship between the period of revolution (P), semimajor axis (D), and the total mass (M) of a binary system: D³ = (M₁ + M₂)P². However, since we are dealing with three stars of equal mass (each equivalent to the mass of our sun), they form a circular orbital pattern. We use the formula P = √((4π²/G)(r³/3M)), where r is the distance from the center of the mass distribution to any of the stars.

Firstly, we must understand that in an equilateral triangle, the distance from the center to a vertex is r = √3/2 * side. So, the distance from the center to any star is 0.866 * 1.0×10¹²m. The gravitational constant, G, is approximately 6.67 * 10^-11 N(m/kg)².

Substituting these values into our formula, we can calculate the period of rotation of the stars. One should note that all three stars have identical masses which simplifies the calculation.

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Answer:

0.545 m

Explanation:

K. E = 26 MeV = 26 x 1.6 x 10^-13 J = 4.16 x 10^-12 J

B = 1.35 T

Let r be the radius of curvature

The formula for the kinetic energy of a cyclotron is given by

[tex]K.E. = \frac{B^{2}q^{2}r^{2}}{2m}[/tex]

m = 1.67 x 10^-27 kg, q = 1.6 x 10^-19 c

[tex]4.16\times 10^{-12} = \frac{1.35^{2}\times (1.6\times 10^{-19})^{2}r^{2}}{2\times 1.67\times 10^{-27}}[/tex]

r = 0.545 m

Explanation:

It is given that, an electron and a proton are separated by a distance of 1.0 m i.e d = 1 m . At this position, F is the force between them

[tex]F=k\dfrac{q_1q_2}{1^2}[/tex]...............(1)

We need to find effect on force between them if the electron moves 0.5 m away from the proton. Let the force is F'.

[tex]F'=k\dfrac{q_1q_2}{0.5^2}[/tex]...............(2)

On dividing equation (1) and (2) we get :

[tex]\dfrac{F}{F'}=\dfrac{k\dfrac{q_1q_2}{1^2}}{k\dfrac{q_1q_2}{0.5^2}}[/tex]

[tex]\dfrac{F}{F'}=0.25[/tex]

F' = 4 F

So, the distance between the electron and the proton is 0.5 m, the new force will be 4 times of previous force.

Final answer:

The electrostatic force between two charges of 1 C each, separated by a distance of 0.5 m can be calculated using Coulomb's Law. The force is inversely proportional to the square of the distance, so if the distance is increased, the force decreases. In this case, if the electron moves 0.5 m away from the proton, the new force can be calculated using Coulomb's Law.

Explanation:

The electrostatic force between two charges of 1 C each, separated by a distance of 0.5 m can be calculated using Coulomb's Law. Coulomb's Law states that the electrostatic force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

So, if the distance between the charges is increased to 1 m, the force between them will decrease. The force is inversely proportional to the square of the distance, which means that if the distance is doubled, the force will decrease by a factor of four.

In this case, if the electron moves 0.5 m away from the proton, the new distance becomes 1.5 m. Using Coulomb's Law, we can calculate the new force:

F = k * (q1 * q2) / r^2

Where F is the force, k is the Coulomb's constant (approximately 9 x 10^9 Nm^2/C^2), q1 and q2 are the charges (in this case, both 1 C), and r is the distance (1.5 m). Plugging in these values, we get:

F = (9 x 10^9 Nm^2/C^2) * (1 C * 1 C) / (1.5 m)^2

F ≈ 4 x 10^9 N

Answer:

The the specific volume and weight are 79.16m³/k mol and 49 N.

Explanation:

Given that,

Diameter = 3 m

Mass of N₂ = 5 kg

We need to calculate the volume of balloon

Using formula of volume

[tex]V= \dfrac{4}{3}\pi r^3[/tex]

[tex]V= \dfrac{4}{3}\pi \times{\dfrac{d}{2}}^3[/tex]

[tex]V= \dfrac{4}{3}\pi \times(\dfrac{3}{2})^3[/tex]

[tex]V=14.137\ m^3[/tex]

We need to calculate the specific volume in the balloon

[tex] Specific\ volume=\dfrac{Volume}{mass}[/tex]

[tex]Specific\ volume=\dfrac{14.137}{5}[/tex]

[tex]Specific\ volume=2.8274\ m^3/kg[/tex]

Now, Molar mass of N₂ gas is 0.028 kg/mol

Now, Specific volume of N₂ gas in the balloon is

[tex]Specific\ volume =0.028\times2.8274=0.0791672\ m^3/mol[/tex]

[tex]Specific\ volume =79.16m^3/kmol[/tex]

(b). We need to calculate the weight of the gas in the balloon

Weight of the balloon is dependent on the mass of the gas.

The weight of the gas is given by

[tex]W = mg[/tex]

[tex]W = 5\times9.8[/tex]

[tex]W = 49\ N[/tex]

Hence, The the specific volume and weight are 79.16m³/k mol and 49 N.

Answer:

Linear momentum, p = 50 kg-m/s

Explanation:

It is given that,

Mass of an object, m = 5 kg

Velocity, v = 10 m/s

We need to find the linear momentum of that object. It is given by :

[tex]p=m\times v[/tex]

[tex]p=5\ kg\times 10\ m/s[/tex]

p = 50 kg-m/s

So, the linear momentum of that object is 50 kg-m/s. Hence, this is the required solution.

The linear momentum of an object with a mass of 5 kg and velocity of 10 m/s is calculated using the formula P = mv, which results in 50 kg*m/s.

The student has asked to calculate the linear momentum of an object that has a mass of 5 kg and a velocity of 10 m/s. To find the linear momentum, we use the formula:

P = mv

Where P is the momentum, m is the mass, and v is the velocity. By plugging in the numbers:

P = 5 kg \\* 10 m/s

P = 50 kg\\*m/s

Therefore, the linear momentum of the object is 50 kg\\*m/s, which corresponds to option (C).

Answer:

one ninth

Explanation:

d = 1 cm , v = v

D = 3d, V = ?

By the equation of continuity,

A V = a v

3.14 x D^2 / 4 x V = 3.14 x d^2 / 4 x v

9d^2 x V = d^2 x v

V = v / 9

Thus, the velocity becomes one ninth the initial velocity

Answer:

The initial speed is 7.174 m/s.

Explanation:

Given that,

Mass = 92.0 kg

Coefficient of friction = 0.61

Time = 1.2 s

We need to calculate the acceleration

Using of friction force

[tex]F = \mu mg[/tex]...(I)

Where, [tex]\mu[/tex] =Coefficient of friction

g = acceleration due to gravity

Using newton's law

[tex]F = ma[/tex]....(II)

m = mass of the baseball

a = acceleration of the baseball

From equation (I) and (II)

[tex]ma=\mu mg[/tex]...(I)

Put the value in the equation

[tex]a=0.61\times9.8[/tex]

[tex]a=5.978\ m/s^2[/tex]

We need to calculate the initial velocity

Using equation of motion

[tex]v = u-at[/tex]

Where, v = final velocity

u = initial velocity

a = acceleration

Here, a is negative because the player comes to rest

t = time

Put the value in the equation

[tex]0=u-5.978\times1.2[/tex]

[tex]u=7.174\ m/s[/tex]

Hence, The initial speed is 7.174 m/s.

The initial speed of the baseball player was approximately 7.176 m/s, calculated using the principles of Physics (Kinematics) and understanding of friction.

The subject of the problem involves a concept in Physics known as Kinematics, specifically dealing with friction and motion. The initial speed of the baseball player can be calculated using the equation of motion: final velocity (v_f) = initial velocity (v_i) + acceleration (a) x time (t), where the final velocity in this case is 0 (as the player comes to rest), time is 1.2 seconds, and the acceleration is the frictional force divided by the mass of the player. Frictional force is obtained by multiplying the mass of the player, the gravitational acceleration, and the coefficient of friction. In this case, substituting acceleration = -0.61 * 9.8 m/s² (because it's a decelerating force), the equation becomes 0 = v_i - 0.61 * 9.8 * 1.2 from which initial velocity (v_i) comes out to be approximately 7.176 m/s.

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Answer:

Correct option is 'd' Surroundings

Explanation:

The space surrounding a thermodynamic system is known as Surroundings

Answer:

37.91594 keV

Explanation:

[tex]E_i[/tex] = Incident energy = 400 keV

θ = 30°

h = Planck's constant = 4.135×10⁻¹⁵ eV s = 6.626×10⁻³⁴ J s

Incident photon wavelength

[tex]\lambda_i=\frac{hc}{E_i}\\\Rightarrow \lambda_i=\frac{4.135\times 10^{-15}\times 3\times 10^8}{400\times 10^3}\\\Rightarrow \lambda_i=3.101\times 10^{-12}\ m[/tex]

Difference in wavelength

[tex]\Delta \lambda=\frac{h}{m_ec}(1-cos\theta)\\\Rightarrow \Delta \lambda=\frac{6.626\times 10^{-34}}{9.11\times 10^{-31}\times 3\times 10^8}(1-cos30)\\\Rightarrow \Delta \lambda=3.248\times 10^{-13}\ m[/tex]

[tex]\lambda_f=\lambda_i+\Delta \lambda\\\Rightarrow \lambda_f=3.101\times 10^{-12}+3.248\times 10^{-13}\\\Rightarrow \lambda_f=3.426\times 10^{-12}[/tex]

Final photon wavelength

[tex]\lambda_f=\frac{hc}{\lambda_f}\\\Rightarrow E_f=\frac{4.135\times 10^{-15}\times 3\times 10^8}{3.426\times 10^{-12}}\\\Rightarrow E_f=362084.06\ eV = 362.08406\ keV[/tex]

Energy of the recoiling electron

[tex]\Delta E=E_i-E_f\\\Rightarrow \Delta E=400-362.08406=37.91594\ keV[/tex]

Energy of the recoiling electron is 37.91594 keV

Answer:

The energy of recoiling electron=192.44 keV

Explanation:

Energy of x-ray[tex]E_o=400 keV[/tex]

Web know that compton shift is

[tex]\Delta \lambda =\dfrac{h}{m_eC(1-cos\theta)}[/tex]

[tex]m_e[/tex] is the mass of electron and C is the velocity of sound.

Given that θ=30°

Now by putting the values

[tex]\Delta \lambda =\dfrac{6.63\times 10^{-34}}{9.11\times 10^{-31}\times 3\times 10^8(1-cos30)}[/tex]

[tex]\Delta \lambda =2.8\times 10^{-3}nm[/tex]

[tex]\Delta \lambda_o=\dfrac{hc}{E_o}[/tex]

By putting the values

[tex]\Delta \lambda_o=\dfrac{6.63\times 10^{-34}3\times 10^8}{400\times 1.602\times 10^{-16}}[/tex]

[tex]\Delta \lambda_o=3.31times 10^{-3}nm[/tex]

[tex]\lambda =\Delta \lambda +\lambda _o[/tex]

[tex]\lambda=2.8\times 10^{-3}+3.31\times 10^{-3}nm[/tex]

[tex]\lambda=6.11`\times 10^{-3}nm[/tex]

Energy [tex]E=\dfrac{hC}{\lambda }[/tex]

So [tex]E=\dfrac{6.63\times 10^{-34}\times 3\times 10^8}{6.11\times 10^{-12}}[/tex]

E=207.55 keV

The energy of recoiling electron=400-207.55 keV

The energy of recoiling electron=192.44 keV

Answer:

443.3 N

567.4 N

Explanation:

Consider the triangle ABC

AC = hypotenuse = magnitude of force vector = F = 720 N

AB = adjacent = Component of force along east-west line = [tex]F_{x}[/tex]

BC = Opposite = Component of force along north-south line = [tex]F_{y}[/tex]

θ = Angle = 38 deg

In triangle ABC

[tex]Sin38 = \frac{BC}{AC}[/tex]

[tex]Sin38 = \frac{F_{y}}{F}[/tex]

[tex]0.616 = \frac{F_{y}}{720}[/tex]

[tex]F_{y}[/tex] = 443.3 N

Also, In triangle ABC

[tex]Cos38 = \frac{AB}{AC}[/tex]

[tex]Cos38 = \frac{F_{x}}{F}[/tex]

[tex]0.788 = \frac{F_{x}}{720}[/tex]

[tex]F_{x}[/tex] = 567.4 N

Answer:

A. Both spheres land at the same time.

Explanation:

The horizontal motion doesn't affect the vertical motion.  Since the two spheres have the same initial vertical velocity and same initial height, they land at the same time.

Answer:

A. Both spheres land at the same time.

Explanation:

If air resistance and friction are ignored, then both spheres land at the same time. Falling objects fall toward the center of the Earth with the same constant acceleration, independent of their mass, when they are defined to be in free-fall. For an object to be in free-fall, it has to be in a situation in which both air resistance and friction are considered negligible. This is true regardless of the direction of the fall.

Answer:

Velocity at the point of maximum x cordinate is 9.578m/s

Position vector of the particle when it reaches point of maximum x ordinate is [tex]\overrightarrow{r}=5.402\widehat{i}+9.76\widehat{j}[/tex]

Explanation:

We shall resolve the motion of the particle along x and y direction separately

The particle will reach it's maximum x coordinate when it's velocity along x axis shall become 0

We have acceleration along x-axis = [tex]-2.6m/s^{2}[/tex]

acceleration along y-axis = [tex]4.7m/s^{2}[/tex]

Thus using the first equation of motion along x axis we get

[tex]v_{x}=u_{x}+a_{x}t\\\\[/tex]

Applying values we get

[tex]0=5.3-2.6t\\\\\therefore t=\frac{5.3}{2.6}sec\\\\t=2.038sec[/tex]

Now to obtain it's position we shall use third equation of motion

[tex]v_{x}^{2}=u_{x}^{2}+2as_{x}\\\\0=(5.3)^{2}+2(-2.6)s_{x}\\\\\therefore s_{x}=\frac{-28.09}{-5.2}m\\\\s_{x}=5.402m[/tex]

Now it's location along y- axis can be obtained using 2nd equation of motion along the y axis

[tex]s_{y}=u_{y}t+\frac{1}{2}a_{y}t^{2}[/tex]

Applying values as follows we get

[tex]u_{y}=0\\a_{y}=4.7m/s^{2}\\t=2.038s[/tex]

[tex]s_{y}=0\times 2.038+\frac{1}{2}\times 4.7m/s^{2}\times2.038^{2}\\\\s_{y}=9.76m[/tex]

thus the position vector of the particle when it reaches it's maximum x co-ordinate is

[tex]\overrightarrow{r}=5.402\widehat{i}+9.76\widehat{j}[/tex]

Now velocity of the particle at the position of maximum x co-ordinate shall be zero along x-axis and along the y-axis it can be found along the first equation of motion along y axis

[tex]v_{y}=u_{y}+a_{y}t\\\\v_{y}=0+4.7\times 2.038\\\\v_{y}=9.578m/s[/tex]

The power drawn by the air conditioner to cool the house from 35°C to 20°C in 38 minutes, given a COP of 2.8, is calculated to be 1.35 kW.

To calculate the power drawn by the air conditioner, we first determine the energy needed to cool the house from 35°C to 20°C. The energy required (Q) can be calculated using the formula Q = m⋅cv⋅ΔT, where m is the mass of the air, cv is the specific heat capacity at constant volume, and ΔT is the change in temperature.

Plugging the given values into the formula: Q = 800 kg ⋅ 0.72 kJ/kg°C ⋅ (35°C - 20°C) = 8,640 kJ.

Next, we use the Coefficient of Performance (COP) of the air conditioner to find the work input (W) required for this cooling process. The formula relating these quantities is COP = Q/W, rearranging for W gives W = Q/COP.

Therefore, W = 8,640 kJ / 2.8 = 3,085.71 kJ. To find the power drawn, we need the work in kilowatts (kW), knowing that 1 kW = 1 kJ/s, and that the total duration is 38 min or 2,280 seconds. Thus, the power drawn is Power = W / time = 3,085.71 kJ / 2,280 s = 1.35 kW.

Power drawn by the air conditioner is approximately 6.76 kW, calculated from COP, heat transfer, and time taken for cooling.

To solve this problem, we can use the energy balance equation for the air within the house:

[tex]\[ Q = mc\Delta T \][/tex]

where:

- [tex]\( Q \)[/tex] is the heat transfer,

- [tex]\( m \)[/tex] is the mass of air,

- [tex]\( c \)[/tex] is the specific heat capacity of air,

- [tex]\( \Delta T \)[/tex] is the change in temperature.

We also know that the Coefficient of Performance (COP) is defined as:

COP = Q_cooling/W_input

where:

- [tex]\( Q_{\text{cooling}} \)[/tex] is the heat removed from the house (in this case),

- W_input is the work input (power consumed) by the air conditioner.

We are given that the COP [tex](\( \text{COP} = 2.8 \))[/tex] and the initial and final temperatures [tex](\( T_{\text{initial}} = 35^\circ C \)[/tex], [tex]\( T_{\text{final}} = 20^\circ C \))[/tex].

First, let's calculate the heat transfer using the energy balance equation:

[tex]\[ Q = mc\Delta T \][/tex]

[tex]\[ Q = (800 \, \text{kg})(1.0 \, \text{kJ/kg}^\circ C)(35 - 20) \][/tex]

[tex]\[ Q = (800 \, \text{kg})(1.0 \, \text{kJ/kg}^\circ C)(15) \][/tex]

Q = 12000kJ

Now, we can find the work input using the COP formula:

COP = Q_cooling/{W_input

W_input = [tex]\frac{Q_{\text{cooling}}}{\text{COP}}[/tex]

W_input = [tex]\frac{12000 \, \text{kJ}}{2.8}[/tex]

W_input ≈ 4285.71 kJ

To find the power drawn by the air conditioner, we need to convert the energy to power. Since the time taken for cooling is 38 minutes, or [tex]\( \frac{38}{60} \)[/tex] hours:

Power = W_input /Time

[tex]\[ \text{Power} = \frac{4285.71 \, \text{kJ}}{\frac{38}{60} \, \text{hours}} \][/tex]

Power ≈ 6762.45 W

So, the power drawn by the air conditioner is approximately [tex]\( 6762.45 \, \text{W} \)[/tex] or [tex]\( 6.76 \, \text{kW} \)[/tex].

To find the speed at the bottom of the ramp, we can use the principles of physics. By equating the potential energy at the top to the kinetic energy at the bottom, we can solve for the velocity at the bottom of the ramp. Using the given values and trigonometry, we can calculate the height of the ramp and then determine the velocity.

To find the speed of the skateboarder at the bottom of the ramp, we can use the principles of physics. Since we are neglecting friction, we can assume that the mechanical energy of the skateboarder is conserved. The mechanical energy consists of both kinetic energy and potential energy. At the top of the ramp, the skateboarder has only potential energy, which is given by the formula: potential energy = mass * gravity * height. At the bottom of the ramp, the skateboarder has only kinetic energy, which is given by the formula: kinetic energy = 1/2 * mass * velocity^2. By equating the potential energy at the top to the kinetic energy at the bottom, we can solve for the velocity at the bottom of the ramp.

In this case, the mass of the skateboarder is not provided, but it cancels out when equating the potential and kinetic energies. So we can assume any mass value as long as it is consistent throughout the calculation. Let's assume the mass of the skateboarder is 1 kg for simplicity. The height of the ramp is not provided, but we can calculate it using trigonometry. The height is the vertical component of the ramp's length, which is given by: height = length * sin(angle). Plug in the values and calculate the height.

Next, we calculate the potential energy at the top of the ramp using the calculated height. The potential energy is then equated to the kinetic energy at the bottom of the ramp and solve for the velocity. Plug in the values and calculate the velocity at the bottom of the ramp.

Answer:

19.06 Volt

Explanation:

r =0.25 m, l = 11 m, B = 0.17 T, f = 13 Hz

length for one turn = 2 x 3.14 x r = 2 x 3.14 x 0.25 = 1.57 m

Total number of turns in 11 m

N = 11 / 1.57 = 7

Peak emf, e0 = N x B x A x 2 x 3.14 x f

e0 = 7 x 0.17 x 3.14 x 0.25 x 0.25 x 2 x 3.14 x 13 = 19.06 Volt

Answer:

a) [tex]V=0.314 m/s[/tex]

b) [tex]\omega=2.09rad/s[/tex]

c) [tex]x(\theta)=0.15\times cos(\frac{2\pi}{3.0} t)[/tex]

Explanation:

Given:

Radius of the circle, r = 15cm = 0.15m

Time period, T  =3.0s

a) The velocity (V) of a particle moving in the circular motion is given as:

[tex]V=\frac{2 \pi r}{T}[/tex]

substituting the given values in the above equation we get

[tex]V=\frac{2\times \pi \times 0.15m}{3.0s}[/tex]

or

[tex]V=0.314 m/s[/tex]

b) Angular speed ([tex]\omega[/tex]) is given as:

[tex]\omega=\frac{2\pi}{T}[/tex]

or

[tex]\omega=\frac{2\times \pi}{3.0s}[/tex]

or

[tex]\omega=2.09rad/s[/tex]

c) The position of the particle on the x-position is given as:

[tex]x(\theta)=rcos(\theta)[/tex]         (reffer the attached figure)

now the relation between the Θ and the time T is given as:

[tex]\omega = \frac{2\pi}{T}=\frac{\theta}{t}[/tex]

or

[tex]\theta= \frac{2\pi}{T}\times t[/tex]

or

[tex]\theta= \frac{2\pi}{3}\times t[/tex]

substituting the values of r and Θ, we get

[tex]x(\theta)=0.15\times cos(\frac{2\pi}{3.0} t)[/tex]

Final answer:

The particle's speed is 0.1π m/s, its angular speed is 2/3π rad/s, and the x-component of the position function is x(t) = 0.15 * cos((2/3π) * t + π).

Explanation:

(a) Speed of the particle: The speed is the distance traveled by a particle along a circular path per unit time. Given a particle completes 1 revolution (which is 2π times the radius r) every 3 seconds, the speed v is given by: v = (2π * 15 cm) / 3 s. Convert 15 cm to meters by dividing by 100 to get 0.15 meters. So, v = (2π * 0.15 m) / 3 s = 0.1π m/s.

(b) Angular speed of the particle: The angular speed ω is the rate of change of the angular displacement and is measured in radians per second. Since the particle completes one revolution every 3 seconds and one revolution is 2π radians, ω = 2π radians / 3 s = 2/3π rad/s.

(c) Position function for the x-component: Assume the particle starts on the negative x-axis, so the angle θ at t=0 is π. The particle moves in a circle with radius r = 0.15 m. The x-component of the position as a function of time t, using the cosine function is x(t) = r * cos(ω*t + π), where ω = 2/3π rad/s. Plugging values, we get x(t) = 0.15 * cos((2/3π) * t + π).

Answer:

(a) 5 hours 20 min

(b)I would have a great inconvenience with the delay of the signal

Explanation:

Hello

To find the time it takes for light to reach Earth from Pluto, we use a rule of three

Step 1

define

distance from sun to earth=x

distance from pluto to earth=40x

Step 2

Rule of three

distance    time

x       ⇒   8 minutes

40x   ⇒   A minutes    

[tex]A=\frac{40x*8*min}{x}\\ A=320 min\\\\to\ convert\ to\ hours\\\\A=320\ min*\frac{1 h}{60 mi}\\ A=5.33 hours\\\\A=5\ hours\ and\ 20min\\\\[/tex]

(a) 5 hours 20 min

Step 3

I would have a great inconvenience with the delay of the signal, since the message I send will arrive 5 hours and 20 minutes later, if the message is answered immediately I will listen to it 320 min after that, more than ten and a half hours to give and receive a hello

Answer:

Centripetal force, F = 9.81 N

Explanation:

It is given that,

Mass of the object (yo-yo), m = 0.12 kg

It is whirled around by its string in a circle

Radius of circle, r = 0.44 m

Speed of object, v = 6 m/s

We need to find the required centripetal force in Newtons to keep this yo -yo whirling in a circle. The centripetal force is given by :

[tex]F=\dfrac{mv^2}{r}[/tex]

[tex]F=\dfrac{0.12\ kg\times (6\ m/s)^2}{0.44\ m}[/tex]

F = 9.81 Newton

So, the centripetal force required to keep this yo -yo whirling in a circle is 9.81 N. Hence, this is the required solution.

Answer:

option (a)

Explanation:

The direction of current is given by the Maxwell's right hand thumb rule.

If we hold a current carrying conductor in our right hand such that the thumb indicates the direction of magnetic field then the curling of fingers give the direction of current.

Here the direction of current is in counter clockwise direction.

Answer:4800

Explanation:

Given data

k=0.4

surface area(A)=100[tex]ft&^{2}[/tex]

Temprature on near side of material[tex](T_1)[/tex]=150°F

temprature on far side of material[tex](T_2)[/tex]=165°F

thickness(t)=1.5in=0.125[tex]ft^2[/tex]

Conductive heat transfer rate(Q) =[tex]\frac{k\times A\times\left ( T_2-T_1\right )}{y}[/tex]

Q=[tex]\frac{0.4\times 100\left ( 165-150\right )}{1.5}[/tex]

Q=4800°F per feet

Answer:

work done by applied force = 1120J

work done by frictional force = 943.25J

work done by the gravitational force = work done by the normal force = 0J

Explanation:

Given:

Mass of the box = 55kg

Displacement of the box = 7.0m across the floor

Applied force = 160N

Coefficient of kinetic friction ([tex]\mu_k[/tex]) = 0.25

Now the work done (W) is given as:

W = F×d

Where,

F = Applied force on the body

d = Displacement of the body in the direction of the applied force

Forces acting on the box are:

1) Applied force

2) Frictional Force

3) Gravitational force

4) Normal force due to the weight

Now the work done by the respective forces:

1) By the applied force

W = 160 kg × 7 m = 1120 J

2) By the frictional force

[tex]W_f =-\mu mg\times displacement[/tex]

Where

g = acceleration due to the gravity

and the negative sign here depicts that the frictional force is acting the against the applied force and the direction of displacement.

thus,

[tex]W_f =-0.25\times 55\times 9.8\times 7[/tex]

or

[tex]W_f =-943.25J[/tex]

The work done by the normal force and the gravitational force will be zero as there is no displacement in the direction of the application of both the forces

The work done by both the gravitational force and normal force is equal to zero (0).

Given the following data:

In this scenario, there are four forces acting on the box and these include:

For the applied force:

Mathematically, the work done due to an applied force is given by this formula:

[tex]Work\;done = force \times distance\\\\Work\;done = 160 \times 7[/tex]

Work done = 1,120 Nm.

For the frictional force:

Mathematically, the work done due to a frictional force is given by this formula:

[tex]Work\;done = -\mu mg \times distance\\\\Work\;done = -0.25 \times 55 \times 9.8 \times 7[/tex]

Work done = -943.25.

However, the work done by both the gravitational force and normal force is equal to zero (0) because there isn't any displacement or distance covered in the direction of the application of these forces.

Read more on work done here: brainly.com/question/22599382