The function​ s(t) represents the position of an object at time t moving along a line. Suppose s (2 )equals 146and s (6 )equals 254.Find the average velocity of the object over the interval of time [2 comma 6 ].

Final answer:

Match the correct statistical methods to their definitions as follows: Frequency counts are A (applied to categorical values), Frequency distribution is B (applied to quantitative values), Measurements of central tendency are C (mean, median, and mode), and Measurements of spread are D (max, min, percentiles, and standard deviation).

Explanation:

When examining statistical methods, it is important to correctly match each concept with its definition to understand data analysis better. Here are the correct matches:

These matches are crucial in the proper analysis and interpretation of data whether you are dealing with categorical or quantitative data. It's important to apply the correct statistical method for the level of measurement being dealt with, such as nominal, ordinal, interval, or ratio scales.

Explanation:

(a)  Formula for average translational kinetic energy of a particle is as follows.

                   U = [tex]\frac{3}{2}(\frac{RT}{N})[/tex]

where,   R = Reydberg's constant

              T = absolute temperature

              N = Avogadro's number

Therefore, we will calculate value of average translational kinetic energy as follows.

                 U = [tex]\frac{3}{2}(\frac{RT}{N})[/tex]

                    = [tex]\frac{3}{2}(\frac{8.314 J/mol K \times 6000 K}{6.023 \times 10^{23} mol^{-1}})[/tex]

                    = [tex]1.24 \times 10^{-19}[/tex] J

Therefore, value of average translational kinetic energy is [tex]1.24 \times 10^{-19}[/tex] J.

(b)   Formula for average kinetic energy is as follows.

             K.E = [tex]\frac{1}{2}(\frac{M}{N})v^{2}_{rms}[/tex]

Here,   M = molar mass = 1 kg/K mol

And, the average kinetic energy is equal to the average translational kinetic energy.

Hence,   K.E = U

     [tex]\frac{1}{2}(\frac{M}{N})v^{2}_{rms}[/tex] = [tex]\frac{3}{2}(\frac{RT}{N})[/tex]

            [tex]v^{2}_{rms} = \frac{3RT}{M}[/tex]

            [tex]v = \sqrt{\frac{3RT}{M}}[/tex]

therefore, we will calculate r.m.s speed of the given atom as follows.

              [tex]v = \sqrt{\frac{3RT}{M}}[/tex]

              [tex]v = \sqrt{\frac{3 \times 8.314 J/mol K \times 6000 K}{1 kg/K mol}}[/tex]

              = 386.84 m/s

Hence, value of r.m.s speed of the given atom is 386.84 m/s.

Following are the solution to the given points:

Given:

Temperature [tex]= 6000\ K[/tex]

To find:

kinetic energy =?

rms speed of the atoms=?

Solution:

For point a)

[tex]\to Avg \ KE =\frac{3}{2}\ KT \\\\[/tex]

[tex]\to KE_{avg} =1.5 \times 1.38 \times 10^{-23} \times 6000 \\\\[/tex]

                [tex]=1.5 \times 1.38 \times 10^{-23} \times 6000 \\\\=2.07 \times 10^{-23} \times 6000 \\\\=2.07 \times 10^{-23} \times 6000 \\\\=12.420 \times 10^{-20}\ J[/tex]

For point b)

[tex]\to[/tex] rms speed [tex]V_{rms}[/tex]:  

[tex]\to v_{rms}=\sqrt{\frac{3RT}{M}}[/tex]

             [tex]=\sqrt{\frac{3 \times 8314 \times 6000}{1}} \\\\=\sqrt{\frac{24942\times 6000}{1}} \\\\=\sqrt{149652000}\\\\=12233.233\\\\ = 12233\ \frac{m}{s}[/tex]

Therefore, the final answer is "[tex]12.420 \times 10^{-20} \ J\ \ and \ \ 12233 \ \frac{m}{s}[/tex]".

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The value of steam iron resistance is [tex]20ohm[/tex].

The relation between voltage, current and resistant shown below,

                      [tex]V=IR\\\\R=\frac{V}{I}[/tex]

Where V is voltage and I is current .

 Given that,  [tex]I=6A,V=1.20*10^{2}[/tex]

Substitute values in above relation.

                  [tex]R=\frac{1.2*10^{2} }{6}=20ohm[/tex]

Hence, the value of steam iron resistance is [tex]20ohm[/tex].

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The resistance of the steam iron can be calculated using Ohm's Law, expressed as the formula R = V/I. With a current of 6.00 A and a voltage of 1.20 ✕ 102 V, the steam iron has a resistance of 20.0 Ohms.

The electrical characteristics of a device, including resistance, can usually be determined using Ohm's Law, which states that the current through a conductor between two points is directly proportional to the voltage across the two points. It can be expressed in the formula R = V/I, where R is resistance, V is voltage, and I is current.

In this case, our steam iron has a current (I) of 6.00 A and a voltage (V) of 1.20 ✕ 102 V. Substituting these values into the Ohm's Law formula gives us: R = (1.20 x 102 V) / 6.00 A.

This calculates to a resistance (R) of 20.0 Ohms for the steam iron.

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Answer:

(1.) gamma rays.    (2.) X-rays.   (3.) ultraviolet.    (4.) visible light.    (5.) infrared.

(6.) radio waves

Explanation:

Notice that these wavelengths span an enormous range. The wavelengths of gamma rays can be smaller than the size of an atomic nucleus, while the wavelengths of radio waves can be many meters (or even milometers) long. Visible light spans only a very narrow range of wavelengths, from about 400 nano meters at the blue (violet) end to about 700 nano meters at the red end.

The types of light in the electromagnetic spectrum, ranked from shortest to longest wavelength, are gamma rays, X-rays, ultraviolet, visible light, infrared, and radio waves.

The electromagnetic spectrum is made up of different types of waves, each characterized by specific wavelengths. The categories of light in the electromagnetic spectrum from shortest to longest wavelength are:

As we move from gamma rays to radio waves, the wavelength increases. Gamma rays have the shortest wavelength, while radio waves have the longest.

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Answer:

Time = 1.36s

Explanation:

coefficient of kinetic friction = μk

μk = F/N = F / mg

where m = mass of the body

g = acceleration due to gravity.

Doubling the force on the box we have,

μk = 2ma/mg

hence μk = 0.15/2 = 0.075

To determine the time it takes to reach a velocity of 2m/s from 1m/s .

From Newton law of motion,

v = u + a*t

for a deceleration of a = μk x g

2m/s = 1m/s + 0.075 x 9.8m/s² x t

t = (2m/s - 1m/s)/0.735m/s²

t = 1.36 seconds..

it will take the 1.36s for the crate to double to 2.0m/s

Answer:3

Explanation:

Convert the distance to meters,the standard units.

Final answer:

When a word problem provides the separation distance between two charges in centimeters, you should (3) convert that distance to meters, the standard SI unit for measurements in physics. Hence, (3) is the correct option.

Explanation:

If a word problem gives the separation distance between two charges in centimeters (cm), you should convert that distance to meters, the standard unit. Physicists use the International System of Units (SI units) for calculations to ensure consistency and comparison of data.

Since the SI unit for displacement is the meter (m), converting from centimeters to meters is necessary for performing electrostatic calculations involving Coulomb's law or any other physics formula.

For instance, when using Coulomb's law to find the force between two charges, the formula is F = k * (q₁ * q₂ / r²), where k is Coulomb's constant (8.987 x 10⁹ N m²/C²), q₁ and q₂ are the charges, and r is the separation distance in meters. If the problem states the distance in centimeters, you would convert the distance to meters by dividing by 100, since there are 100 centimeters in a meter.

Answer:

13.125 m

Explanation:

b) The quantity which are known as

Initial velocity of life preserver, u = 1.40 m/s

time to reach, t = 1.50 s

final velocity of the life preserver, v = 0 m/s

acceleration due to gravity = 9.8 m/s²

a) height of the preserver above water

   using equation of motion

    [tex]h = u t + \dfrac{1}{2}gt^2[/tex]

    [tex]h = 1.4\times 1.50 + \dfrac{1}{2}\times 9.8\times 1.5^2[/tex]

          h = 13.125 m

Hence, the height from where preserver is thrown is equal to 13.125 m

Final answer:

The life preserver was released from approximately 8.925 meters above the water. This calculation was based on the initial velocity, time it took to reach the water, and the acceleration due to gravity, making use of kinematic equations.

Explanation:

To find out how high above the water the life preserver was released, we can use the kinematic equations for uniformly accelerated motion. Since there is an initial velocity and the acceleration due to gravity is applicable, the following kinematic equation can be used:

h = vot + ½at²

Where:

Plugging in the known values into the kinematic equation, we get:

h = (1.40 m/s)(1.50 s) + ½(-9.8 m/s2)(1.50 s)²

Calculating this gives us:

h = 2.10 m - 11.025 m = -8.925 m

Since height can't be negative, we take the absolute value which is 8.925 m. Hence, the life preserver was released approximately 8.925 m above the water.

Knowns

Answer:

10.12square feet

Explanation:

Pressure exerted on the object is defined as the ratio of the force exerted on it to its unit area. Mathematically, Pressure = Force/Area

Given the force = 170kN

Pressure = 180KPa

Area = Force /Pressure

Area = 170kN/180KPa

Area = 0.94N/Pa

Note that 1Newton/Pascal = 10.764square feet

Therefore 0.94N/Pa = x

x = 0.94× 10.764

x = 10.12square feet

Therefore the area upon which this pressure is exerted is 10.12sqft.

Answer:

F= 12 N

Explanation:

Given that

Work done by student ,W= 60 J

The mass of the box ,m = 3 kg

Length ,x = 5 m

We know that ,The work done by a force a force F is given as

W= F .x

x=Displacement

F=Force

W=Work

Now by putting the values

60 = F x 5

[tex]F=\dfrac{60}{5}\ N[/tex]

F= 12 N

That is why the magnitude of the force will be 12 N.

The magnitude of the force applied to the box to do the work is 12 N

Workdone is defined as the product of force and distance moved in the direction of the force. Mathematically, it can be expressed as:

Workdone (Wd) = force (F) × distance (d)

Wd = Fd

With the above formula, we can obtain the force used to do the work.

•Workdone (Wd) = 60 J

•Distance (d) = 5 m

•Force (F) =?

Wd = Fd

60 = F × 5

Divide both side by 5

F = 60 / 5

F = 12 N

Thus, 12 N is required to do the work.

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Answer: 200ohms

Explanation:

According to Ohm's law, the current (I) passing through a metallic conductor is directly proportional to the potential difference (v) between its ends at constant temperature and pressure. Mathematically, E=IRt

Where E is the voltage across both bulbs, I is the current, Rt is the total equivalent resistance

E = 100V, I = 0.25amps Rt = R+R(since they are 2 identical bulbs in series, we will add them together)

Rt = 2R

Substituting this values in the formula to get R, we have;

100 = 0.25(2R)

100 = 0.5R

R = 100/0.5

R = 200ohms

The resistance of a single light bulb will be 200ohms.

Note that the unit of resistance is ohms

Answer:

Neural processing

Explanation:

Neural processing- it is referred to as the unit that is responsible for the implementation of arithmetic logic which is crucial for the execution of machine learning algorithms.

it is used to operate the artificial neural network in designing and can be helpful to enhance the efficiency of machine learning applications like artificial language.

Answer:

a) 0.069 acres

b) 0.114acres

c) 17.78acres

Explanation:

1 dot=1/36sqin

a) 1 sqin= 330×330=108900ft^2

1 dot=108900/36 =3025ft^2

Converting to acre,divide by43560

3025/43560 =0.069acre

b) 1 chain =22 yards

25×22=550yards

Converting to acre divide by 4840

550/4840 =0.114acre

c)1 sqmile =640acres

(1/36) ÷ 640

640/36

17.78acres

To calculate the area represented by each dot on the dot grid, you need to consider the scale of the map. With a scale of 330 ft. per inch, 1 dot represents (330^2 / 43,560) acres. With a scale of 25 chains per inch, 1 dot represents ((25 chains)^2 / 10) acres. And with a scale of 1 mile per inch, 1 dot represents ((1 mile)^2 / 640) acres.

Explanation:

The conversion from dots to acres varies depending on the scale:

Answer: relative motion between observer and the sound source.

Explanation: The Doppler effect states that when there is a relative motion between an observer and a sound source the frequency of sound perceived by the observer is different in frequency from the original from the source.

The mathematical back up for this claim is given below.

f' = (v+v') /(v-vs) × f

Where f' = observed frequency

v = speed of sound in air

v' = velocity of observer

vs = velocity of source

f = frequency of sound source.

From the formulae, it can be seen that a change in the value of the velocity of observer (v') and source (vs) produces different value of observed frequency (f').

Note, frequency of sound (f) is a constant.

Answer: Relative motion

Explanation: If two objects are moving either towards or away from each other with both having their velocities in a reference frame and someone is outside this reference frame seeing the motion of the two objects.

The observer ( in his own frame of reference) will measure a different velocity as opposed to the velocities of the two object in their own reference frame. p

Both the velocity measured by the observer in his own reference frame and the velocity of both object in their reference is correct.

Velocities of this nature that have varying values based on motion referenced to another body is known as relative velocity.

Motion of this nature is known as relative motion.

Note that the word reference frame is simply any where the motion is occurring and the specified laws of motion is valid

For this example of ours, the reference frame of the companion is the train and the telephone poles has their reference frame as the earth.

The companion will measure the velocity of the telephone poles relative to him and the velocity of the telephone pole relative to an observer outside the train will be of a different value.

Answer:q=3.536+10^-6C

Explanation:

Answer:

[tex]r=61.65m[/tex]

Explanation:

Since the package remains in contact with the car's seat, the package's speed is equal to the car's speed. At the top on the mountain the package's centripetal force must be equal to its weight:

[tex]mg=F_c[/tex]

The centripetal force is defined as:

[tex]F_c=ma_c=\frac{mv^2}{r}[/tex]

Here v is the linear speed of the object and r is the radius of curvature. We need to convert the linear speed to [tex]\frac{m}{s}[/tex]:

[tex]88.5\frac{km}{h}*\frac{1000m}{1km}*\frac{1h}{3600s}=24.58\frac{m}{s}[/tex]

Now, we calculate r:

[tex]mg=\frac{mv^2}{r}\\r=\frac{v^2}{g}\\r=\frac{(24.58\frac{m}{s})^2}{9.8\frac{m}{s^2}}\\\\r=61.65m[/tex]

Answer:

12.24 m/s

Explanation:

Speed: This can be defined as the rate of change of distance with time. The S.I unit of speed is m/s.

Using the formula,

a = v/t................ Equation 1

Where a = acceleration of the sprinter, v = speed of the sprinter, t = time.

making v the subject of the equation,

v = at ................. Equation 2

Given: a = 5.1 m/s², t = 2.4 s.

Substitute into equation 2

v = 5.1(2.4)

v = 12.24 m/s.

Hence, the speed of the sprinter = 12.24 m/s

Answer:

Explanation:

given that

Radius =0.75m

Cnet=0.13nC

a. Electric field inside the sphere located 0.5m from the center of the sphere.

The electric field located inside the sphere is zero.

b. The electric field located 0.25m beneath the sphere.

Since the radius is 0.75m

Then, the total distance of the electric field from the centre of the circle is 0.75+0.25=1m

Then

E=kq/r2

K=9e9Nm2/C2

q=0.13e-9C

r=1m

Then,

E= 9e9×0.13e-9/1^2

E=1.17N/C. Q.E.D

Answer:

Magnitude of Electric field E at at point 0.50m which is within the sphere is Zero( i.e E = 0)

Explanation:

It is understand from Guass' law, the electric field in a region enclosed by a conducting sphere is Zero.

It is given that the radius of the sphere is 0.75m, therefore, a point located at 0.50m from the sphere centre 0.25m before the sphere surface still falls inside the sphere, therefore making the electric field at that point zero in magnitude.

To find the mass attached to a spring when the frequency of oscillation is known, use the formula for the frequency of SHM (f = (1/2π) ∙ √(k/m)) and rearrange it to solve for mass.

To determine the mass attached to a horizontal spring where the frequency of oscillation is known, we can use the formula for the frequency of a mass-spring system undergoing simple harmonic motion (SHM), which is f = (1/2π) ∙ √(k/m), where f is the frequency, k is the spring constant, and m is the mass. Given a spring constant 9 N/m and a frequency 0.9 Hz, we can rearrange the formula to solve for the mass (m = k / (2πf)^2). Plugging in the values, we get the mass m = 9 N/m / (2π ∙ 0.9 Hz)^2, which we can calculate to find the mass of the object attached to the spring.

Using the frequency equation for a mass-spring system, the mass can be determined to be approximately 0.28 kg.

To find the mass attached to the spring, we can use the equation for the frequency of a mass-spring system:

Where:

This equation can be rearranged to solve for mass:

Plugging in the provided values:

First, simplify the expression inside the brackets:

Then:

Thus, the mass attached to the spring is approximately 0.28 kg.

To find the approximate value of the effective spring stiffness of the interatomic bond, calculate the stiffness constant of the titanium wire using Young's modulus.

To find the approximate value of the effective spring stiffness of the interatomic bond, we need to calculate the stiffness constant of the titanium wire. The stiffness constant, or Young's modulus (Y), is given by the formula Y = F/A/L, where F is the force applied, A is the cross-sectional area of the wire, and L is the original length of the wire.

First, we need to find the force applied. The force can be calculated using the equation F = mg, where m is the mass and g is the acceleration due to gravity.

Next, we need to calculate the cross-sectional area of the wire. The cross-sectional area can be calculated using the formula A = π(r^2), where r is the radius of the wire.

Finally, we can substitute the values into the formula Y = F/A/L to find the Young's modulus of the titanium wire.

The metadata is intended to help a search feature the least likely feature contained in the metadata is An archive containing previous versions of the e-book so, option A is correct.

The term "e-book" refers to a digital file with a body of text and graphics that is suited for electronic distribution and displays on-screen similarly to a printed book.

E-books can be made by converting a printer's source files into forms that are simple to download and read on a screen, or they can be taken from a database or a collection of text files that weren't made just for printing.

When businesses like Peanut Press started offering book content for reading on personal digital assistants (PDAs), handheld devices that were the forerunners of today's smartphones and tablet computers, the market for buying and selling e-books first became a mainstream industry in the late 1990s.

Thus, metadata does not contain a feature to find an archive containing previous versions of the e-book.

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The potential energy stored in the spring when the car is pushed against it is most nearly is 0.026 J.

Non linear spring is

[tex]F = -kx^2 \rightarrow force[/tex]

The potential energy

[tex]\frac{dU}{dX} = -F\\\\U = \int\limits {-Fdx}\\\\U = + \int\limits^x_0 {kx^2dx}\\\\U = \frac{kx^3}{3} \\\\U = 5000 \div 3 (\frac{2.5}{100}^)2\\\\U = \frac{5000 \times 2.5\times 2.5\times 2.5}{3\times 100\times 100\times 100}[/tex]

= 0.026 J

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Answer:15 watts

Explanation:

To get the current

Power P=iV

(Where P is power of bulb 60watts, i is current, and V is the voltage 120V

i=P/V

i = 60/120

i = 1/2A

 But

V=iR where R is the resiance

Therefore R=V/i

                       =120÷1/2

                        =120×2

                         =240ohms

For series connection, each bulb draws the same current since,

Since the two bulbs are 60-watt bulbs, they will have the same resistance, therefore the voltage across each bulb is the same and equals halve of the applied voltage, 120/2 = 60 volts.

Taken the resistance 240 ohm is constant across the series, we can  roughly estimate the current flow and calculate power dissipation in the series connection.

We have 120 V/(240 + 240 ) ohms = 1/4 A.

The power dissipated in each bulb is (1/4) × 60 = 15 watts.

When two 60 W bulbs are connected in series across a 120 V outlet, each bulb consumes 30 W of power because they share the same current, and the voltage is divided equally among them due to their series connection.

When two 60 W bulbs are connected in series across a 120 V outlet, they share the same current because they are in a series circuit. In a series circuit, the current is constant throughout, and the voltage is divided among the components. Since power is related to both voltage and current, we can use the formula P = VI to find out how much power each bulb consumes.

Total Resistance (R_total): In a series circuit, the total resistance is the sum of the individual resistances. Let's assume the resistance of each bulb is R.

Since P = VI, we can rearrange the formula to find current (I):

I = P / V

The power of each bulb is 60 W, and the voltage is 120 V, so the current passing through the circuit is:

I = 60 W / 120 V = 0.5 A

Voltage Across Each Bulb: In a series circuit, the voltage is divided among the components. Each bulb receives a portion of the total voltage. So, the voltage across each bulb is 60 V.

Power Consumed by Each Bulb: Now, we can use the formula P = VI to find the power consumed by each bulb. We know the voltage (60 V) and the current (0.5 A) passing through each bulb:

P = (0.5 A) * (60 V) = 30 W

Each bulb consumes 30 W of power when connected in series across the 120 V outlet.

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Answer:

Explanation:

Given

Combined mass of cab and elevator is [tex]M=1510\ kg[/tex]

acceleration of coin with respect to elevator [tex]a_{ce}=-7.4\ m/s^2[/tex]

[tex]a_{ce}[/tex] can be written as

[tex]a_{ce}=a_{cg}-a_{eg}[/tex]

where

[tex]a_{cg}[/tex]=acceleration of coin with respect to ground

[tex]a_{eg}[/tex]=acceleration of elevator with respect to ground

[tex]a_{ce}=a_{cg}-a_{eg}[/tex]

[tex]-7.4=-9.8-a_{eg}[/tex]

[tex]a_{eg}=-9.8+7.4=-2.4\ m/s^2[/tex]

Forces on elevator

[tex]T-Mg=Ma_{eg}[/tex]

[tex]T=M(g+a_{eg})[/tex]

[tex]T=1510\times (9.8-2.4)[/tex]

[tex]T=11,174\ N[/tex]

Here we considered downward direction as negative and upward as positive

Answer:

1. Burning coal contributes to sulfur aerosols in the stratosphere; these sulfur aerosols have moderated global warming.

Explanation:

Sulfur aerosols both absorb as well as scatter solar radiation coming from the sun , thus saving the earth from getting wormed up . In this way it has moderating effect on the global warming . This is also called global dimming effect. Less of the solar radiation is able to reach earth surface due to these aerosol.

Answer:

A. describes how an object accelerates when a force is applied

Explanation:

Newton's second law of motion concerns the behavior of objects that do not have a stability between all established forces. The second law states that an object's acceleration depends on two factors: the net force on the entity and the entity's mass.

Answer:

The velocity of watermelon when it passes Superman is 78 m/s.

Explanation:

Height of the building, d = 320 m

Speed of the superman, v = 39 m/s

We need to find the speed of watermelon when it passes Superman. Let t is the time taken by the watermelon. So,

[tex]d=ut+\dfrac{1}{2}gt^2[/tex]

Here u = 0

[tex]d=\dfrac{1}{2}gt^2[/tex]

[tex]39t=\dfrac{1}{2}\times 9.8\times t^2[/tex]

t = 7.95 seconds

Let v is the speed of the watermelon. It is given by :

[tex]v=gt[/tex]

[tex]v=9.8\times 7.95[/tex]

v = 77.91 m/s

or

v = 78 m/s

So, the velocity of watermelon when it passes Superman is 78 m/s.

Answer:

The equation that will relate all the given parameters, in other to calculate the potential energy of charge A is:

∆V = ∆U/q, ∆V is potential at charge A position, q is magnitude of charge A, ∆U will be made the subject of the relation, which is the Potential Energy of charge A. The notation "∆" show, the quantities have both in values and final values, in the electric field.(Change in Electric potential and potential energy, due to the effect of the field)

Explanation:

The potential energy of a charged particle (Charge A) in an electric field depends on the magnitude of the charge(Known as stated in the question). However, the potential energy per unit charge has a unique value at any point in the electric field.

Answer:

0.26087 rad/s

Explanation:

mass of the child (m) = 40 kg

velocity (v) = 3 m/s

distance (r) = 1.5 m

moment of inertia (I) = 600 kg.m^{2}

rotational momentum of the child = Iω

where

rotational momentum of the child = Iω = 90 x 2 = 180 kg[tex]m^{2}[/tex]/s

from the conservation of momentum the initial momentum of the child must be the same as the final momentum of the child

initial momentum of the child = final momentum of the child

180 = (90 + 600) ω

180 = 690 ω

ω = 180 / 690 = 0.26087 rad/s

The angular or rotational momentum is the conserved quantity that has both magnitude and direction. It is given by:

[tex]\rm L = mvr[/tex]

Where,

L = rotational/angular momentum, m = mass, v = velocity and r = radius

The angular speed will be 0.26087 rad/s.

The speed can be estimated as:

Given,

Rotational momentum is calculated by Iω.

Where,

[tex]\begin{aligned}&= 40 \times (1.5)^{2} \\&= 90 \rm kg/m^{2}\end{aligned}[/tex]

[tex]\begin{aligned}&= \dfrac{3 }{1.5} \\&= 2 \rm rad/s\end{aligned}[/tex]

Rotational momentum of the child = Iω

[tex]\begin{aligned} \rm I\omega &= 90 \times 2 \\\\&= 180 \;\rm kgm^{2}/s\end{aligned}[/tex]

The final and the initial momentum would be the same according to the conservation law.

Initial momentum = Final momentum

180 = (90 + 600) ω

180 = 690 ω

Solving further:

[tex]\begin{aligned}\omega &= \dfrac{180}{690}\\\\&= 0.26087 \;\rm rad/s\end{aligned}[/tex]

Therefore, 0.26087 rad/s is the angular speed.

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Explanation:

The given data is as follows.

        Initial temperature ([tex]T_{1}[/tex]) = [tex]40^{o}F[/tex] = 499.67R

        Initial pressure ([tex]P_{1}[/tex]) = 16.8 Psi

        Volume (V) = 5 [tex]ft^{3}[/tex]

       Work done = 4 Btu

        Final temperature ([tex]T_{2}[/tex]) = 90 F

So,  R of [tex]O_{2}[/tex] = 0.3353 Psi ft^{3}/lbm R

Therefore, we will calculate the mass of oxygen as follows.

            M = [tex]\frac{PV}{RT}[/tex]

                = [tex]\frac{16.8 \times 5}{0.3353 \times 499.67}[/tex]

                = 0.50137 lbm

Therefore, mass of oxygen is 0.50137 lbm.

Now, we will calculate the change in internal energy as follows.

          [tex]\Delta U = mC_{v} [T_{2} - T_{1}][/tex]

                       = [tex]0.50137 \times 0.157 \times (90 - 40)[/tex]

                       = 3.936 BTU

Relation between heat energy and internal energy is as follows.

           [tex]\Delta Q = \Delta u + W.d[/tex]

                      = [tex]3.936 + (-4)[/tex]

                      = -0.0642 BTU

Therefore, amount of heat transfer in BTU is 0.0642 BTU.