Calculate the wavelengths of the first five members of the lyman series of spectral lines

Answer:

a) V = 0.085 m^3

b) m = 17 kg

Explanation:

1) Data given

mb = 68 kg (mass for the block)

20% of the block volume is floating

100-20= 80% of the block volume is submerged

2) Notation

mb= mass of the block

Vw= volume submerged

mw = mass water displaced

V= total volume for the block

3) Forces involved (part a)

For this case we have two forces the buoyant force (B), defined as the weight of water displaced acting upward and the weight acting downward (W)

Since we have an equilibrium system we can set the forces equal. By definition the buoyant force is given by :

B = (mass water displaced) g = (mw) g   (1)

The definition of density is :

[tex] \rho_w = \frac{m_w}{V_w} [/tex]

If we solve for mw we got [tex]m_w = \rho_w V_w [/tex]  (2)

Replacing equation (2) into equation (1) we got:

[tex] B = \rho_w V_w g [/tex] (3)

On this case Vw represent the volume of water displaced = 0.8 V

If we replace the values into equation (3) we have

0.8 ρ_w V g = mg  (4)

And solving for V we have

 V =  (mg)/(0.8 ρ_w g )

We cancel the g in the numerator and the denominator we got

V = (m)/(0.8 ρ_w)

V = 68kg /(0.8 x 1000 kg/m^3) = 0.085 m^3

4) Forces involved (part b)

For this case we have bricks above the block, and we want the maximum mass for the bricks without causing  it to sink below the water surface.

We can begin finding the weight of the water displaced when the block is just about to sink (W1)

W1 = ρ_w V g

W1 = 1000 kg/m^3 x 0.085 m^3 x 9.8 m/s^2 = 833 N

After this we can calculate the weight of water displaced before putting the bricks above (W2)

W2 = 0.8 x 833 N = 666.4 N

So the difference between W1 and W2 would represent the weight that can be added with the bricks (W3)

W3 = W1 -W2 = 833-666.4 N = 166.6 N

And finding the mass fro the definition of weight we have

m3 = (166.6 N)/(9.8 m/s^2) = 17 Kg

Final answer:

The volume of the block of wood and the maximum mass of bricks it can carry without sinking are determined using principles of buoyancy and uniform density.

Explanation:

(a) To determine the volume of the block of wood, we use the fact that 20.0% of its volume is above the water. Given a uniform density, this means the submerged volume is 80% of the total volume. Hence, the volume of the block is 0.8 x (68.0 kg / 920 kg/m³) = 0.472 m³.

(b) The maximum mass of bricks that can be loaded without sinking the block can be calculated using the concept of buoyancy. The buoyant force equals the weight of the water displaced, so the mass of the bricks should not exceed the mass of the water displaced by the volume of the submerged part of the block. Therefore, the maximum mass of bricks is 0.472 m³ x 1100 kg/m³ x 9.8 m/s² = 5148.64 kg.

Answer:

[tex]\rho = 0.50 g/L[/tex]

Explanation:

As we know that

PV = nRT

here we have

[tex]P = 1.0 atm[/tex]

[tex]P = 1.013 \times 10^5 Pa[/tex]

so we have

[tex]V = 1.4 \times 10^{-3} m^3[/tex]

T = 290 K

now we have

[tex](1.013 \times 10^5)(1.4 \times 10^{-3}) = n(8.31)(290)[/tex]

[tex]n = 0.06 [/tex]

now the mass of gas is given as

[tex]m = n M[/tex]

[tex]m = (0.06)(28)[/tex]

[tex]m = 1.65 g[/tex]

now density of gas when its volume is increased to 3.3 L

so we will have

[tex]\rho = \frac{m}{V}[/tex]

[tex]\rho = \frac{1.65 g}{3.3 L}[/tex]

[tex]\rho = 0.50 g/L[/tex]

Final answer:

The density of a gas can be calculated using the ideal gas law, which states that PV = nRT. To find the density, we can rearrange the equation and substitute the given values.

Explanation:

The density of a gas can be calculated using the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. To find the density, we can rearrange the equation as follows:

Density = mass/volume = (molar mass * n) / V

Using the given information, we can substitute the values into the equation and solve for the density.

Answer:

the angular velocity increase by a factor of 2

Explanation:

using the law of the conservation of the angular momentum

[tex]L_i = L_f[/tex]

where [tex]L_i[/tex] is the inicial angular momentum and [tex]L_f[/tex] is the final angular momentum.

also, the angular momentum can be calculated by:

L = IW

where I is the inertia momentum and the W is the angular velocity.

so:

[tex]I_i W_i = I_fW_f[/tex]

we know that [tex]I_f = \frac{1}{2}I_i[/tex] then,

[tex]I_iW_i = \frac{1}{2}I_iW_f[/tex]

solving for [tex]W_f[/tex]:

[tex]W_f = 2W_i[/tex]

When a figure skater pulls her arms in while spinning, her moment of inertia decreases and her angular speed increases. In the provided case, with the moment of inertia decreasing by half, the angular speed will double.

When a figure skater is spinning with her arms outstretched, and she pulls her arms in close to her body, her moment of inertia decreases. According to the law of conservation of angular momentum, if the moment of inertia of a spinning object decreases, its angular speed must increase to keep the angular momentum constant. In this case, since the skater's moment of inertia decreases by half, her angular speed will correspondingly double, or increase by a factor of 2.

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Answer:

The speed in the smaller section is [tex]43.2\,\frac{m}{s}[/tex]

Explanation:

Assuming all the parts of the pipe are at the same height, we can use continuity equation for incompressible fluids:

[tex] \Delta Q=0 [/tex] (1)

With Q the flux of water that is [tex] Av[/tex] with A the cross section area and v the velocity, so by (1):

[tex] A_{2}v_{2}-A_{1}v_{1}=0 [/tex]

subscript 2 is for the smaller section and 1 for the larger section, solving for [tex] v_{2} [/tex]:

[tex]v_{2}=\frac{A_{1}v_{1}}{A_{2}} [/tex] (2)

The cross section areas of the pipe are:

[tex] A_{1}=\frac{\pi}{4}d_{1}^{2} [/tex]

[tex] A_{2}=\frac{\pi}{4}d_{2}^{2} [/tex]

but the problem states that the diameter decreases 86% so [tex] d_{2}=0.86d_{1} [/tex], using this on (2):

[tex] v_{2}=\frac{\frac{\pi}{4}d_{1}^{2}v_{1}}{\frac{\pi}{4}d_{2}^{2}}=\frac{\cancel{\frac{\pi}{4}d_{1}^{2}}v_{1}}{\cancel{\frac{\pi}{4}}(0.86\cancel{d_{1}})^{2}}\approx1.35v_{1} [/tex]

[tex]v_{2}\approx(1.35)(32)\approx43.2\,\frac{m}{s} [/tex]

Answer:

(A) Vf = 13.8 m/s

(B)  Vf = 15.1 m/s      

Explanation:

length of rope (L) = 35 m

angle to the vertical = 44 degrees

acceleration due to gravity (g) = 9.8 m/s^{2}

(A) from conservation of energy

final kinetic energy + final potential energy = initial kinetic energy + initial potential energy

0.5m(Vf)^{2} + mg(Hf) =  0.5m(Vi)^{2} + mg(Hi)

where

m = mass

Hi = initial height = 35 cos 44 = 25.17

Hf = final height = length of vine = 35 m

Vi = initial velocity = 0 since he starts from rest

Vf = final velocity

the equation now becomes

0.5m(Vf)^{2} + mg(Hf) = mg(Hi)

0.5m(Vf)^{2} = mg (Hi - Hf)

0.5(Vf)^{2} = g (Hi - Hf)

0.5(Vf)^{2} = 9.8 x (25.17 - 35)

0.5(Vf)^{2} = - 96.3  (the negative sign tells us the direction of motion is downwards)

Vf = 13.8 m/s

(B) when the initial velocity is 6 m/s the equation remains

      0.5m(Vf)^{2} + mg(Hf) =  0.5m(Vi)^{2} + mg(Hi)

       m(0.5(Vf)^{2} + g(Hf)) =  m(0.5(Vi)^{2} + g(Hi))

      0.5(Vf)^{2} + g(Hf) = 0.5(Vi)^{2} + g(Hi)

      0.5(Vf)^{2} = 0.5(Vi)^{2} + g(Hi) - g(Hf)

       0.5(Vf)^{2} = 0.5(6)^{2} + (9.8 x (25.17 - 35))

        0.5(Vf)^{2} =  -114.3  ( just as above, the negative sign tells us the direction of motion is downwards)      

       Vf = 15.1 m/s

Answer:

a) [tex]v_{f} \approx 0.328\,\frac{m}{s}[/tex], b) [tex]v_{f} \approx 6.009\,\frac{m}{s}[/tex]

Explanation:

Let consider that bottom has a height of zero. The motion of Tarzan can be modelled after the Principle of Energy Conservation:

[tex]U_{g,1} + K_{1} = U_{g,2} + K_{2}[/tex]

The final speed is:

[tex]K_{2} = U_{g,1} - U_{g,2} + K_{1}[/tex]

[tex]\frac{1}{2}\cdot m \cdot v_{f}^{2} = m\cdot g \cdot L\cdot (\cos \theta_{2}-\cos \theta_{1}) + \frac{1}{2}\cdot m \cdot v_{o}^{2}[/tex]

[tex]v_{f}^{2} = 2 \cdot g \cdot L \cdot (\cos \theta_{2} - \cos \theta_{1}) + v_{o}^{2}[/tex]

[tex]v_{f} = \sqrt{v_{o}^{2}+2\cdot g \cdot L \cdot (\cos \theta_{2}-\cos \theta_{1})}[/tex]

a) The final speed is:

[tex]v_{f} = \sqrt{(0\,\frac{m}{s} )^{2}+2\cdot (9.807\,\frac{m}{s^{2}} )\cdot (35\,m)\cdot (\cos 0^{\textdegree}-\cos 44^{\textdegree})}[/tex]

[tex]v_{f} \approx 0.328\,\frac{m}{s}[/tex]

b) The final speed is:

[tex]v_{f} = \sqrt{(6\,\frac{m}{s} )^{2}+2\cdot (9.807\,\frac{m}{s^{2}} )\cdot (35\,m)\cdot (\cos 0^{\textdegree}-\cos 44^{\textdegree})}[/tex]

[tex]v_{f} \approx 6.009\,\frac{m}{s}[/tex]

Answer:

5295.3 N

Explanation:

According to law of momentum conservation, the change in momentum of the ball shall be from the momentum generated by the batter force

mv + P = mV

P = mV - mv = m(V - v)

Since the velocity of the ball before and after is in opposite direction, one of them is negative

P = 0.14(44.8 - (-19.5)) = 9 kg m/s

Hence the force exerted to generate such momentum within 1.7ms (0.0017s) is

F = P/t = 9/0.0017 = 5295.3 N

Final answer:

The escape velocity of a spherical asteroid with a given radius and given gravitational acceleration can be calculated using a special formula. The distance a particle travels from the surface when it leaves it at a given radial velocity can be determined using the projectile height equation. The speed at which an object hits an asteroid after falling from a given height can be calculated using the falling object terminal velocity equation.

Explanation:

(a) Escape velocity can be defined as the minimum speed required for an object to escape the gravitational pull of a celestial body. To calculate the escape velocity on a spherical asteroid we can use the formula:

escape velocity = sqrt(2 * acceleration due to gravity * radius).

Using the given values, the escape velocity of a spherical asteroid is:

exhaust rate = sqrt(2 * 3.00 m/s2 * 500 000 m) = 6928 m/s.

(b) To calculate the distance from the surface that a particle will travel when it leaves the surface of the asteroid with a radial velocity of 1000 m/s, we can use the formula for the height of the projectile:

Height = (radial velocity)2 / (2 * acceleration due to gravity).

After entering the specified values, the particle travels the distance:

Altezza = (1.000 m/s)2 / (2 * 3,00 m/s2) = 166.666,67 m.

(c) To calculate the speed at which an object hits an asteroid as it falls 1000 km above the surface, we can use the equation for the final velocity of the falling object:

Final velocity = sqrt (initial velocity2 + 2 * acceleration due to gravity * height).

Substituting the given values, the object hits the asteroid with a speed of:

Final velocity = square(0 + 2 * 3.00 m/s2 * 1,000,000 m) = square(6,000,000 m2/s2) = 2,449 m/s.

Answer:

option C

Explanation:

Let mass of the bullet be m and velocity be v

mass of gun be M and bullet be V

now,

using conservation of momentum for gun 1

(M+m) V' = 2 mv + 3 MV

V' = 0

3 M V = - 2 mv

momentum of gun 1 =- 2 mv---------(1)

now for gun 2

(M+m) V' = mv + MV

V' = 0

M V = - mv

momentum of gun 1 = -mv-----------(2)

dividing equation (1) by (2)

[tex]\dfrac{P_m1}{P_m2} = \dfrac{- 2mv}{-mv}[/tex]

[tex]\dfrac{P_m1}{P_m2} = \dfrac{2}{1}[/tex]

the correct answer is option C

The correct option is Option C (2:1).The ratio of the momentum imparted to gun #1 to that imparted to gun #2 is 2:1. This conclusion follows from the principle of conservation of momentum. Thus, the correct answer is option c) 2:1.

Define the mass of the bullet from gun #2 as m. The bullet from gun #1 then has a mass of 2m since it is twice as heavy. Denote the velocity of both bullets as v:

So, the momentum imparted to gun #1 is 2m * v, and for gun #2 it is m * v.

To find the ratio of the momentum imparted to gun #1 to that imparted to gun #2, we calculate:

Therefore, the correct answer is option c) 2:1

Answer:

a) B

b) N

c) B

d) B

e) N

Explanation:

a) The matter in our bodies is the regular matter, basically, we are made of carbon molecules and water. So it involves baryonic matter.

b) In this case, the weakly interacting subatomic particles know as (WIMPs), is the primary candidate for dark matter and this kind of particle has not yet been discovered. We are talking about the nonbaryonic matter.

c) Planets as a Jupiter are made of baryonic matter, in the specific case of Jupiter, it is approximately 75% hydrogen and 24% helium by mass and they are baryonic matter.

d) By definition, brown dwarfs are objects which have a size between a giant gaseous planet like Jupiter and a smaller star, so using the definitions above they are made of baryonic matter.

e) The majority of dark matter is made of non-baryonic matter.

Answer:

Increase in temperature will be [tex]0.389^{\circ}C[/tex]

Explanation:

We have given mass of the aluminium m = 0.095 kg

Height h = 55 m

Specific heat of aluminium c = 900 J/kg°C

We know that potential energy is given as

[tex]PE=mgh=0.095\times 9.8\times 55=51.205[/tex]

Now 65 % of potential energy [tex]=\frac{51.205\times 65}{100}=33.28[/tex]

Now this energy is used to increase the temperature

So [tex]mc\Delta T=33.28[/tex]

[tex]0.095\times 900\times \Delta T=33.28[/tex]

[tex]0.095\times 900\times \Delta T=33=0.389^{\circ}C[/tex]

Answer:

Explanation:

Kinetic energy of the block

= 1/2 m v²

= .5 x 3 x 4 x 4

= 24 J

Negative work of - 24 J is required to be done on this object to bring it to rest.

magnitude of acceleration due to frictional force

= force / mass

2 / 3

= 0 .67 m /s²

Let the body slide by distance d before coming to rest so work done by force = Kinetic energy

=  2 x d = 24

d = 12 m

The magnitude of the work done to bring the block to rest is 2 times the distance the block slides. The magnitude of the acceleration of the block is 2/3 m/s^2. The block slides a distance of 8/3 m before it comes to rest.

The work done on an object is given by the equation:

Work = Force x Distance

In this case, the work done on the block to bring it to rest is equal to the force applied multiplied by the distance the block slides.

Given that the force exerted by the surface is 2 Newtons, we can calculate the magnitude of the work done:

Work = 2 N x Distance

To determine the distance the block slides, we need to calculate its deceleration using Newton's second law:

Force = mass x acceleration

Since the friction force is constant and in the opposite direction of motion, we can write:

2 N = 3 kg x acceleration

Solving for acceleration, we find:

acceleration = 2 N / 3 kg

With the acceleration calculated, we can use the kinematic equation:

vf^2 = vi^2 + 2ad

Since the final velocity is 0 (block comes to rest), the equation simplifies to:

0 = (4 m/s)^2 + 2(-acceleration)d

Solving for distance, we find:

d = (4 m/s)^2 / (2 x -acceleration)

Now, we can substitute the calculated acceleration into the equation to find the distance the block slides.

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Answer:

t = 0.0735 m

Explanation:

Angular acceleration of the flywheel is given as

[tex]\alpha = 3 rad/s^2[/tex]

now after t = 8 s the speed of the flywheel is given as

[tex]\omega = \alpha t[/tex]

[tex]\omega = 3 \times 8 [/tex]

[tex]\omega = 24 rad/s[/tex]

now rotational kinetic energy of the wheel is given as

[tex]K = \frac{1}{2}I\omega^2[/tex]

[tex]K = \frac{1}{2}(\frac{1}{2}mR^2)(24^2)[/tex]

[tex]800 = \frac{1}{4}m(0.23)^2(24^2)[/tex]

[tex]m = 105 kg[/tex]

now we have

[tex]m = \rho (\pi R^2) t[/tex]

[tex]105 = 8600(\pi \times 0.23^2) t[/tex]

[tex]t = 0.0735 m[/tex]

We have that for the Question, it can be said that  thickness must it have to store 800 J of kinetic energy at t = 8.00 s

h=0.0735m

From the question we are told

Generally the equation for the   is mathematically given as

[tex]N=\pir^2h*P\\\\N=3.14*(23*10^{-2}^2)*h*8600\\\\N=1428.5h\\\\[/tex]

Therefore

[tex]E=\frac{1}{2}Iw^2\\\\800=\frac{1}{2}*(\frac{NR^2}{2}w^2)\\\\800=\frac{1}{2}*(\frac{(1428.523*10^{-2})^2}{2}(2*8)^2)[/tex]

h=0.0735m

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Answer:

 θ₂ = 3 θ₁

Explanation:

given,

telescope of lens diameter = 12 inch

another telescope of lens diameter = 4 inch

comparison of resolution power.

Using the formula of resolution

 [tex]\theta = \dfrac{1.22 \lambda}{D}[/tex]

for diameter = 12 inch

 [tex]\theta_1 = \dfrac{1.22 \lambda}{D_1}[/tex].....(1)

for diameter = 4 inch

 [tex]\theta_2 = \dfrac{1.22 \lambda}{D_2}[/tex].......(2)

dividing equation (2) from (1)

[tex]\dfrac{\theta_2}{\theta_1} = \dfrac{D_1}{D_2}[/tex]

now,

[tex]\dfrac{\theta_2}{\theta_1} = \dfrac{12}{4}[/tex]

[tex]\dfrac{\theta_2}{\theta_1} =3[/tex]

 θ₂ = 3 θ₁

hence, we can say that resolution of telescope of 12 inch is 3 time smaller than the resolution of 4 inch telescope.

Answer: Thermal comductivity (K) is 3.964x 10 ^-3 W/m.k

Explanation:

Thermal comductivity K = QL/A∆T

Q= Amount of heat transferred through the material in watts = 75W

L= Distance between two isothermal planes = 0.740mm

A= Area of the surface in square metres = 2m^2

∆T= Temperature change = (37-30) °C.

Solving this : K =( 75 x 0.740 x 10^-3)/ 2 x (37-30)

K = 3.964x 10 ^-3 W/m.k

The question pertains to calculating the thermal conductivity of human skin, a Physics concept linked to heat transfer. By using Fourier's Law of heat conduction, and rearranging the formula to solve for thermal conductivity using given data, an approximate thermal conductivity can be obtained.

The query is related to the determination of thermal conductivity of human skin based on the known parameters. This phenonmenon belongs to the domain of Physics, specifically heat transfer. Here, thermal conductivity is the measure of a material's ability to conduct heat. In this scenario, you have to consider the heat conduction through the skin, which relies on Fourier's Law of heat conduction. It can be represented as:

Q = (k*A*(T1 - T2))/d

Where, Q is the heat transfer rate, k is the thermal conductivity, A is the area of heat transfer, T1 and T2 are the initial and final temperatures, and d is the thickness of the material, in this case, the skin.

From the given data, you can plug in the values into this formula. However, our primary objective is to find out 'k'. Rearranging the formula to find k gives us:

k = (Q * d) / (A * (T1 - T2))

Now, if we put all the given values into the formula, we get:

k = (75 W * 0.00074 m) / (2 m^2 * (37°C - 30°C))

Solving this would provide us with the estimate of thermal conductivity of the skin.

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Answer:

option 1

Explanation:

given,                                    

Weight of crate = 100 N              

Crate is lifted up to height = 1.5 m

Work done =?                    

work = Force  x distance              

work = 100 N    x 1.5 m            

work = 150 J                                        

However, work would be more than the 150 J to lift the crate slightly higher to its final position.

The correct answer is option 1

Answer

Given,

Flow rate of river is equal to 300,000 L/s.

Width of river = 18 m

and depth of river = 22 m

a) Average speed of river

     Q = 300,000 L/s

         = 300 m³/s

     Q = Av

     [tex]v = \dfrac{Q}{A}[/tex]

     [tex]v = \dfrac{300}{18 \times 22}[/tex]

     [tex]v = \dfrac{300}{396}[/tex]

     [tex]v = 0.757\ m/s[/tex]

b) Average speed when river is widen to 63 m and depth is increased to

     [tex]v = \dfrac{Q}{A}[/tex]

     [tex]v = \dfrac{300}{63 \times 42}[/tex]

     [tex]v = \dfrac{300}{2646}[/tex]

     [tex]v = 0.113\ m/s[/tex]

Final answer:

The average speed of the river in the gorge is 0.75 m/s, and downstream of the falls, when the river widens and its depth increases, the average speed decreases to 0.125 m/s.

Explanation:

To calculate the average speed of the river at Huka Falls, we can use the formula for flow rate, which is the volume of fluid passing a point in the river per unit of time:

Flow rate (Q) = Area (A)  imes Velocity (V)

(a) Average speed in the gorge: Given that the flow rate (Q) is 300,000 liters per second (which is equal to 300 cubic meters per second, since 1,000 liters is equal to 1 cubic meter), and the cross-sectional area of the river in the gorge (A) is 20 meters wide  imes 20 meters deep (400 square meters), we can solve for velocity (V) using the formula:

Q = A  imes V

300 m³/s = 400 m²  imes V

V = 300 m³/s \/ 400 m²

V = 0.75 meters per second

(b) Average speed downstream of the falls: Downstream, the river widens to 60 meters and deepens to an average of 40 meters, so the cross-sectional area is 2,400 square meters. Using the same flow rate, we can find the new velocity:

Q = A  imes V

300 m³/s = 2,400 m²  imes V

V = 300 m³/s \/ 2,400 m²

V = 0.125 meters per second

Answer:

The gravitational force is [tex]1.05\times10^{-8}\ N[/tex]

Explanation:

Given that,

Mass of large ball = 8.4 kg

Mass of small ball = 0.061 kg

Separation = 0.057 m

Gravitational constant [tex]G= 6.67\times10^{-11}\ Nm^2/kg^2[/tex]

We need to calculate the gravitational force

Using formula of gravitational force

[tex]F= \dfrac{Gm_{1}m_{2}}{r^2}[/tex]

Put the value into the formula

[tex]F=\dfrac{6.67259\times10^{-11}\times8.4\times0.061}{(0.057)^2}[/tex]

[tex]F=1.05\times10^{-8}\ N[/tex]

Hence,  The gravitational force is [tex]1.05\times10^{-8}\ N[/tex]

To find the gravitational force between two masses, you use the formula derived from Newton's law of universal gravitation, F = G × (m1 × m2) / r², and with the provided values, the force is calculated to be approximately 1.19 × 10⁻¹° Newtons.

The student has asked about the gravitational force between two masses using the apparatus similar to the one used in the Cavendish experiment. To calculate the magnitude of the gravitational force between the large lead balls (8.4 kg each) and the small balls (0.061 kg each), separated by a distance of 0.057 m, and using the universal gravitational constant (G = 6.67259 × 10⁻¹¹ Nm²/kg²), the following formula derived from Newton's law of universal gravitation is used:

F = G × (m1 × m2) / r²

Substituting the given values:

F = (6.67259 × 10⁻¹¹) × (8.4 × 0.061) / (0.057²)

After performing the calculation, we find that the gravitational force F is approximately 1.19 × 10⁻¹° Newtons. This force is a direct application of the universal law of gravitation, indicating that two masses will always exert a gravitational pull on each other, no matter how small.

The mass of Actinium 227 left is 0.0078 kg

Explanation:

The amount of mass left of a radioactive isotope after time t is given by the equation:

[tex]m(t) = m_0 (\frac{1}{2})^{-\frac{t}{\tau_{1/2}}}[/tex]

where

[tex]m_0[/tex] is the initial amount of the sample

t is the time

[tex]\tau_{\frac{1}{2}}[/tex] is the half-life of the isotope

For the sample of Actinium 227 in this problem,

[tex]m_0 = 2 kg[/tex]

[tex]\tau_{1/2}=20 years[/tex]

t = 160 years

Substituting into the equation,

[tex]m(160) = (2 kg) (\frac{1}{2})^{-\frac{160}{20}}=0.0078 kg[/tex]

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Answer:

1) [tex]\Delta L= 0.612\ m[/tex]

2) a. [tex]\Delta V_G=0.57\ L[/tex]

   b. [tex]\Delta V_S=0.021\ L[/tex]

   c. [tex]V_0=0.549\ L[/tex]

Explanation:

1)

∴Change in temperature:

[tex]\Delta T=T_f-T_i[/tex]

[tex]\Delta T=25-(-15)[/tex]

We have the equation for change in length as:

[tex]\Delta L= L.\alpha. \Delta T[/tex]

[tex]\Delta L= 1275\times 12\times 10^{-6}\times 40[/tex]

[tex]\Delta L= 0.612\ m[/tex]

2)

Given relation:

[tex]\Delta V=V.\beta.\Delta T[/tex]

where:

[tex]\Delta V[/tex]= change in volume

V= initial volume

[tex]\Delta T[/tex]=change in temperature

a)

final temperature of gasoline, [tex]T_{Gf}=25^{\circ}C[/tex]

∴Change in temperature of gasoline,

[tex]\Delta T_G=T_{Gf}-T_{Gi}[/tex]

[tex]\Delta T_G=25-15[/tex]

[tex]\Delta T_G=10^{\circ}C[/tex]

Now,

[tex]\Delta V_G= V_G.\beta_G.\Delta T_G[/tex]

[tex]\Delta V_G=60\times 950\times 10^{-6}\times 10[/tex]

[tex]\Delta V_G=0.57\ L[/tex]

b)

final temperature of tank, [tex]T_{Sf}=25^{\circ}C[/tex]

∴Change in temperature of tank,

[tex]\Delta T_S=T_{Sf}-T_{Si}[/tex]

[tex]\Delta T_S=25-15[/tex]

[tex]\Delta T_S=10^{\circ}C[/tex]

Now,

[tex]\Delta V_S= V_S.\beta_S.\Delta T_S[/tex]

[tex]\Delta V_S=60\times 35\times 10^{-6}\times 10[/tex]

[tex]\Delta V_S=0.021\ L[/tex]

c)

Quantity of gasoline spilled after the given temperature change:

[tex]V_0=\Delta V_G-\Delta V_S[/tex]

[tex]V_0=0.57-0.021[/tex]

[tex]V_0=0.549\ L[/tex]

Explanation:

A radio galaxy is a galaxy with a powerful radio luminance relative to the stars'  visible and infrared luminosity. Radio galaxies can emit strong and  speedy particle jets. They inject in their surroundings high amounts of kinetic energy. Many radio galaxies have jets of  plasma shooting out in opposite direction. Thus they ionize the gases surrounding them.

The change in internal energy of the system is +187 J

Explanation:

According to the first law of thermodynamics, the change in internal energy of a system is given by the equation:

[tex]\Delta U = Q + W[/tex]

where

[tex]\Delta U[/tex] is the change in internal energy

Q is the heat absorbed by the system

W is the work done on the system

For the system in this problem, we have

W = +108 J is the work done on it

Q = +79 J is the hear added to it

So, the change in internal energy is

[tex]\Delta U = 108 + 79 = +187 J[/tex]

Learn more about thermodynamics here:

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Answer:

V = 100.8 V

Explanation:

given,

hum being can be electrocuted with current = 48 mA = 0.048 A

Resistance of the man = 2100 Ω

Fatal voltage  = ?

we know,

V = I R

V is the fatal voltage in Volts

R is the resistance provided by the human body

I is current

V = I R

V = 0.048 x 2100

V = 100.8 V

the voltage which can be considered as fatal is equal to  V = 100.8 V

Final answer:

To find the two possible frequencies of the untuned piano string, we can use the formula for beat frequency. From the given information, the original frequency of the untuned string can be either 266.0 Hz or 262.0 Hz. To find the percentage increase in tension on the untuned string, we can use the formula for calculating percentage increase.

Explanation:

To find the two possible frequencies of the untuned piano string, we can use the formula for beat frequency:

Beat frequency = |Frequency of the first string - Frequency of the second string|

In this case, the beat frequency is given as 2.00 s. The frequency of the first string is 264.0 Hz. Let's assume the frequency of the second string is x Hz.

So, we can set up the equation:

2.00 = |264.0 - x|

Solving for x, we get two possible frequencies: 266.0 Hz and 262.0 Hz.

To find the original frequency of the untuned string, we can use the formula:

Original frequency = Frequency of the first string ± Beat frequency

For positive beat frequencies, the original frequency would be:

Original frequency = 264.0 + 2.00 = 266.0 Hz

For negative beat frequencies, the original frequency would be:

Original frequency = 264.0 - 2.00 = 262.0 Hz

To find the percentage increase in tension on the untuned string, we can use the formula:

Percentage increase = (Change in tension / Original tension) x 100

Since the tension on the first string is unchanged (as it is the tuned string), the change in tension on the untuned string is equal to the change in frequency. Assuming the original frequency of the untuned string is 262.0 Hz:

Change in tension = |Original frequency - New frequency|

Change in tension = |262.0 - 264.0| = 2.0 Hz

Therefore, the percentage increase in tension on the untuned string is:

(2.0 / 262.0) x 100 = 0.763%

Answer:

4-6 millimeters

Explanation:

Global warming is causing devastating consequences for the planet such as rising sea levels and temperature in the oceans.

The melting of glaciers is one of the main causes of sea level rise.

Undoubtedly, the most affected and vulnerable areas correspond to areas of Asia, Africa and South America. Specifically, the study highlights that in six Asian countries such as China, Bangladesh, India, Vietnam, Indonesia and Thailand, there are approximately 237 million people who will suffer these floods if defense mechanisms are not activated.

To find the final angular velocity of the system, we need to apply the principle of conservation of angular momentum. The final angular momentum can be obtained by equating the initial angular momentum and the final angular momentum of the system. Solving for the final angular velocity gives us a value of approximately 38.54 rad/s.

To find the final angular velocity of the system, we need to apply the principle of conservation of angular momentum. The initial angular momentum of the system is given by:

Li = Itωt + Irωr

Where It and Ir are the moments of inertia of the turntable and the runner respectively, and ωt and ωr are their respective angular velocities.

Since the runner comes to rest relative to the turntable, ωr = 0. Therefore, the final angular momentum of the system is:

Lf = Itωt

Using the conservation of angular momentum principle, we can set Li equal to Lf:

Itωt = Itωt

Substituting the given values:

81.0kg × m² × 0.190rad/s = It × ωt

Solving for ωt, we find that the final angular velocity of the system is approximately 38.54 rad/s.

To find the final angular velocity of the runner and turntable system, we apply conservation of angular momentum. The final angular velocity is calculated to be 0.611 rad/s. This involves determining the initial angular momenta of both the runner and the turntable, then using the total moment of inertia to find the final velocity.

To find the final angular velocity of the system when the runner comes to rest relative to the turntable, we need to apply the principle of conservation of angular momentum. The initial angular momentum of the system (runner plus turntable) must equal the final angular momentum since there are no external torques acting.

Step-by-Step Calculation :

Determine the initial angular momentum of the runner:

Determine the initial angular momentum of the turntable:

Calculate the total initial angular momentum:

Find the final angular velocity:

The final angular velocity of the system is 0.611 rad/s.

The backbone of the DNA molecule is formed by alternating sugar and phosphate groups. The nitrogenous bases are located in the interior of the molecule.

The backbone of the DNA molecule is made up of the alternating sugar and phosphate groups. The sugar and phosphate groups are bonded by covalent bonds, and they line up on the outside of each strand. The nitrogenous bases, which include adenine (A), thymine (T), cytosine (C), and guanine (G), are stacked in the interior of the DNA molecule.

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The backbone of a DNA molecule is formed from alternating sugar and phosphate groups of nucleotides. The nitrogenous bases, which are not part of the backbone, protrude from it and are involved in the formation of the double helix structure.

The backbone of the DNA molecule consists of alternating sugar and phosphate groups. A nucleotide, which is the building block of DNA, consists of three components: a nitrogenous base, a pentose sugar, and a phosphate group. The backbone is formed by the bonding of the phosphate group of one nucleotide to the sugar of the next nucleotide, creating a chain of sugar-phosphate bonds.

For instance, the phosphate group is attached to the 5' carbon of one nucleotide and the 3' carbon of the next nucleotide. In this sense, the DNA molecule can be visualized as a twisted ladder where the backbone represents the rails of the ladder and the nitrogenous bases represent the steps.

It's important to note that the nitrogenous bases are not part of the backbone; They stick out from the backbone and are involved in hydrogen bonding with the nitrogenous bases of the complementary DNA strand, resulting in a double helix structure. The two strands of the DNA run in opposite directions making them antiparallel.

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Answer:

112.06 kg - Thats heavy !

Explanation:

Let's do force balance here. Let the object of our interest be George. The forces acting on him are the tension in  the upward direction, his weight in the downward direction and the centrifugal force in the downward direction. Considering the upward and downward directions on the y-axis and f=given the fact that George doesn't move up or down, the forces are balanced along the y-axis. Hence doing force balance:

magnitude of forces upward =magnitude of forces downward

i.e., Tension(T) = Weight(mg) + Centrifugal force (mv²/r)

where: 'm' is the mass of George, g is the acceleration due to gravity (9.8 m/s²). v is the speed with which George moves (14.1 m/s) and r is the radius of the circle in which he's moving at the instant (Here since he's swinging on the rope, he moves in a circle with radius as the length of the rope and hence r=7.3m).

therefore, T = m (9.8 + (14.1)²/7.3) = 4150 N

Therefore, m = 112.06 kg

Answer:

955.36 seconds ≈ 16 minutes

Explanation:

Power(P) is the rate of doing work(W)

That is, P = W/t, where t is the time.

multipying both sides with 't' and dividing with 'P', we get: t=W/P

Here, W = 5.35 x 10^10 J and P = 5.6 x 10^7 W ( 1 W = 1 J/s).

Therefore , on dividing W with P, we get 955.36 seconds.

Explanation:

Here velocity of ball and bat are in opposite direction.

Velocity of ball = 40 m/s

Velocity of bat = 18 m/s

Since they are in opposite direction relative velocity is given by,

           Relative velocity = 40 + 18 = 58 m/s

Distance to home plate = 18 m

We have

                Displacement = Velocity x Time

                           18 = 58 x Time

                          Time = 0.3 seconds

Option A is the correct answer.