A piece of charcoal used for cooking is found at the remains of an ancient campsite. A 1.09 kg sample of carbon from the wood has an activity of 2020 decays per minute. Find the age of the charcoal. Living material has an activity of 15 decays/minute per gram of carbon present and the half-life of 14C is 5730 y. Answer in units of y.

To calculate the coefficient of friction between the cheese and ice, we can use the work-energy theorem. By calculating the work done on the cheese by the friction force, we can determine the coefficient of friction. By plugging in the given values, the coefficient of friction is found to be 0.047.

To find the coefficient of friction between the cheese and ice, we can use the concept of work-energy theorem. The work done on the cheese by the friction force is equal to the change in kinetic energy of the cheese. The work done by friction is given by the equation:

Work = Force x Distance

In this case, the force is the friction force and the distance is the distance the cheese slid. We can express the friction force as:

Friction Force = coefficient of friction x Normal Force

Since the cheese is in contact with the ice, the normal force exerted on the cheese is equal to its weight:

Normal Force = mass of cheese x acceleration due to gravity

Substituting the expressions for friction force and normal force into the work equation, we get:

Work = (coefficient of friction x mass of cheese x acceleration due to gravity) x distance

Since the work done is equal to the change in kinetic energy of the cheese, we have:

0.5 x mass of cheese x final velocity^2 - 0.5 x mass of cheese x initial velocity^2 = (coefficient of friction x mass of cheese x acceleration due to gravity) x distance

Simplifying the equation, we can solve for the coefficient of friction:

coefficient of friction = (0.5 x mass of cheese x (final velocity^2 - initial velocity^2)) / (mass of cheese x acceleration due to gravity x distance)

Plugging in the given values, we find that the coefficient of friction between the cheese and ice is 0.047.

The answer is  [tex]\mu \[/tex]approx 0.0153.

The coefficient of friction between the cheese and the ice can be determined by analyzing the conservation of momentum and the work-energy principle.

First, let's calculate the initial momentum of the pellet:

[tex]\[ p_{pellet} = m_{pellet} \cdot v_{pellet} \][/tex]

[tex]\[ p_{pellet} = 0.0013 \, \text{kg} \cdot 73 \, \text{m/s} \][/tex]

[tex]\[ p_{pellet} = 0.0949 \, \text{kg} \cdot \text{m/s} \][/tex]

When the pellet gets stuck in the cheese, the momentum is transferred to the cheese-pellet system. By conservation of momentum:

[tex]\[ m_{pellet} \cdot v_{pellet} = (m_{pellet} + m_{cheese}) \cdot v_{cheese} \][/tex]

[tex]\[ 0.0013 \, \text{kg} \cdot 73 \, \text{m/s} = (0.0013 \, \text{kg} + 0.109 \, \text{kg}) \cdot v_{cheese} \][/tex]

[tex]\[ v_{cheese} = \frac{0.0013 \, \text{kg} \cdot 73 \, \text{m/s}}{0.1103 \, \text{kg}} \][/tex]

[tex]\[ v_{cheese} = \frac{0.0949 \, \text{kg} \cdot \text{m/s}}{0.1103 \, \text{kg}} \][/tex]

[tex]\[ v_{cheese} \approx 0.8608 \, \text{m/s} \][/tex]

Now, we use the work-energy principle to find the coefficient of friction. The work done by friction is equal to the kinetic energy lost by the cheese as it slides to a stop:

[tex]\[ W_{friction} = f \cdot d \][/tex]

[tex]\[ f = \mu \cdot N \][/tex]

[tex]\[ N = m_{cheese} \cdot g \][/tex]

[tex]\[ W_{friction} = \mu \cdot m_{cheese} \cdot g \cdot d \][/tex]

The kinetic energy lost by the cheese is equal to its initial kinetic energy:

[tex]\[ KE_{initial} = \frac{1}{2} m_{cheese} \cdot v_{cheese}^2 \][/tex]

[tex]\[ KE_{initial} = \frac{1}{2} \cdot 0.109 \, \text{kg} \cdot (0.8608 \, \text{m/s})^2 \][/tex]

[tex]\[ KE_{initial} \approx 0.0396 \, \text{kg} \cdot \text{m}^2/\text{s}^2 \][/tex]

[tex]\[ KE_{initial} \approx 0.0396 \, \text{J} \][/tex]

Setting the work done by friction equal to the kinetic energy lost:

[tex]\[ \mu \cdot m_{cheese} \cdot g \cdot d = KE_{initial} \][/tex]

[tex]\[ \mu \cdot 0.109 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 \cdot 0.25 \[/tex], [tex]\text{m} = 0.0396 \, \text{J} \][/tex]

[tex]\[ \mu \cdot 2.5963 \, \text{kg} \cdot \text{m/s}^2 = 0.0396 \[/tex], [tex]\text{J} \][/tex]

[tex]\[ \mu \approx \frac{0.0396 \, \text{J}}{2.5963 \, \text{kg} \cdot \text{m/s}^2} \] \[ \mu \approx 0.0153 \][/tex]

 The correct format for the answer is:

[tex]\[ \boxed{\mu \approx 0.0153} \][/tex]

Answer:

The correct answer is the speed accuracy decreases

Explanation:

Before examining the final statements, let's review the uncertainty principle

       Δx  Δp> = h ’/ 2

       h ’= h / 2π

This is because the process of measuring one quantity affects the measurement of the other.

Let's review the claims

The speed of the particle is proportional to the moment

           p = mv

Therefore, if the position is measured more accurately (x) the accuracy of p must decrease

        Δp = h ’/ 2 Δx

The correct answer is the speed accuracy decreases

According to the Heisenberg uncertainty principle in physics, if the accuracy in measuring a particle's position increases, the accuracy in measuring its velocity decreases. This is a fundamental characteristic of the quantum world due to the wave-particle duality of matter.

The question refers to the Heisenberg uncertainty principle in physics, particularly quantum mechanics. This principle states that the more precisely the position of a particle is known, the less precisely its velocity (or momentum, to be more specific) can be known, and vice versa. Therefore, if the accuracy of measuring a particle's position increases, then the accuracy in measuring its velocity decreases.

This phenomenon isn't due to measurement techniques or technology. It's a fundamental limit defined by the nature of the quantum world. It derives from the wave-particle duality of matter, which means small particles like electrons behave both as particles and waves. If we measure the position very precisely, the particle acts more like a wave with an uncertain speed. Conversely, if we measure the speed very precisely, the particle acts more like a particle with an uncertain position.

For example, if we use an extremely short-wavelength electron probe to measure an electron's position, we'd be very accurate but massively disturb the electron's velocity in the process. Hence, increasing the precision in position measurement increases the uncertainty in velocity measurement.

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The amplitude of Maxwell's displacement current flowing across the gap between the plates of the capacitor is 1.753 x 10^-7 A.

The amplitude of Maxwell's displacement current flowing across the gap between the plates of this capacitor can be determined using the formula for the displacement current, which is given by ID = ε0AdV/dt, where ε0 is the permittivity of free space, A is the area of the capacitor plates, and dV/dt is the rate of change of voltage with respect to time.

In this case, the area A of the circular plates is equal to πa^2, where a is the radius of the plates. Therefore, A = π(0.01 m)^2 = 0.000314 m^2. The rate of change of voltage can be obtained from the equation V(t) = V0sin(2πft), where V0 is the amplitude of the voltage, f is the frequency, and t is the time.

Substituting the given values V0 = 10 V and f = 100 Hz into the equation, we have V(t) = 10sin(2π(100)t). Differentiating this equation with respect to time gives dV/dt = 2000πcos(2π(100)t). Plugging in the value of t = 0 (for maximum displacement current) into the equation, we find that cos(2π(100)(0)) = cos(0) = 1.

Therefore, the amplitude of the Maxwell's displacement current ID is given by ID = (8.85 x 10^-12 F/m)(0.000314 m^2)(2000π)(1) = 1.753 x 10^-7 A.

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Answer

The answer and procedures of the exercise are attached in the following archives.

Step-by-step explanation:

You will find the procedures, formulas or necessary explanations in the archive attached below. If you have any question ask and I will aclare your doubts kindly.  

Final answer:

To explore how the incorporation of tungsten carbide (WC) into cobalt (Co) affects the composite material's modulus of elasticity, the moduli values are calculated for a range of WC volume percentages under both isostrain and isostress conditions, illustrating the performance envelope and guiding material design in engineering applications.

Explanation:

To plot the modulus of elasticity vs. the volume percent of WC in Co from 0 to 100 vol%, we use both upper- and lower-bound expressions to form a performance envelope. For the isostrain (upper-bound) case, the mixture's modulus of elasticity (Ecomposite, isostrain) can be calculated using the rule of mixtures formula: Ecomposite, isostrain = ECoVCo + EWCVWC, where E represents modulus of elasticity, and V represents volume fraction. For the isostress (lower-bound) case, the inverse of the composite's modulus of elasticity can be expressed as 1/Ecomposite, isostress = VCo/ECo + VWC/EWC. Here, ECo = 200 GPa is the modulus of elasticity for cobalt, and EWC = 700 GPa is for tungsten carbide.

As the volume percent of WC increases from 0 to 100%, the calculated Ecomposite, isostrain values will show a trend of increasing modulus due to the higher modulus of WC compared to Co. Conversely, the Ecomposite, isostress values will also follow an upward trend but at a different rate, illustrating the material's variation within the performance envelope depending on the volume fraction of WC. This graphical representation will help to visualize how incorporating tungsten carbide into cobalt affects the composite's mechanical performance, underlining the importance of material design and engineering applications.

Answer:

The temperature change of the copper is greater than the temperature change of the water.

Explanation:

deltaQ = mc(deltaT)

Where,

delta T = change in the temperature

m =mass

c = heat capacity

[tex]\frac{(deltaT)_{Cu}}{(deltaT)_{w}}=\frac{4190J/kg.K}{390J/kg.K}\\ \\(deltaT)_{Cu}=10.74(deltaT)_{w}[/tex]

The temperature change in the copper is nearly 11 times the temperature change in the water.

So, the correct option is,

The temperature change of the copper is greater than the temperature change of the water.

Hope this helps!

The temperature change of the copper is equal to the temperature change of the water.

The temperature change of the copper is equal to the temperature change of the water.

When two substances come into contact, heat is transferred between them until they reach thermal equilibrium, where their temperatures are equal. According to the Law of Conservation of Energy, the heat lost by one substance is equal to the heat gained by the other substance. In this case, since all heat transfer occurs between the copper and water, the temperature change of the copper is equal to the temperature change of the water.

Answer:

0.00493 m/s

Explanation:

T = Temperature of the isotope = 85 nK

R = Gas constant = 8.341 J/mol K

M = Molar mass of isotope = 86.91 g/mol

Root Mean Square speed is given by

[tex]v_r=\sqrt{\dfrac{3RT}{M}}\\\Rightarrow v_r=\sqrt{\dfrac{3\times 8.314\times 85\times 10^{-9}}{86.91\times 10^{-3}}}\\\Rightarrow v_r=0.00493\ m/s[/tex]

The Root Mean Square speed is 0.00493 m/s

Final answer:

The RMS speed of Rubidium-87 atoms at 85 nK can be estimated using the ideal gas approximation and the formula 'Urms = √(3kBT/M)'. The molar mass is converted to kg/mol and the temperature to kelvins before calculation.

Explanation:

To calculate the root-mean-square (RMS speed) of Rubidium-87 atoms at a temperature of 85.0 nK assuming classical ideal gas behavior, we can use the formula:

Urms = √(3kBT/M)

Where Urms is the root-mean-square speed, kB is Boltzmann's constant (1.38 × 10-23 J/K), T is the absolute temperature in kelvins, and M is the molar mass in kilograms per mole. First, we convert the molar mass of Rubidium-87 from grams per mole to kilograms per mole by dividing it by 1000:

M = 86.91 g/mol ∖ 0.08691 kg/mol

Next, we convert the temperature from nanokelvins to kelvins:

T = 85.0 nK = 85.0 × 10-9 K

Substituting the values into the RMS speed equation gives us:

Urms = √(3 × (1.38 × 10-23 J/K) × (85.0 × 10-9 K) / 0.08691 kg/mol)

After calculating, we find that the RMS speed of Rubidium-87 atoms at 85.0 nK is:

Urms ≈ ... m/s

Note that the calculation here assumes that the Rubidium atoms behave as a classical ideal gas, which is not an accurate assumption for atoms at such low temperatures.

Answer:

a. [tex]3.95\times10^{26} [/tex]W

Explanation:

[tex]T[/tex] = temperature of the surface of sun = 5800 K

[tex]r[/tex] = Radius of the Sun = 7 x 10⁸ m

[tex]A[/tex] = Surface area of the Sun

Surface area of the sun is given as

[tex]A = 4\pi r^{2} \\A = 4(3.14) (7\times10^{8})^{2}\\A = 6.2\times10^{18} m^{2}[/tex]

[tex]e[/tex] = Emissivity = 1

[tex]\sigma[/tex] = Stefan's constant = 5.67 x 10⁻⁸ Wm⁻²K⁻⁴

Using Stefan's law, Power output of the sun is given as

[tex]P = \sigma e AT^{4} \\P = (5.67\times10^{-8}) (1) (6.2\times10^{18}) (5800)^{4}\\P = 3.95\times10^{26} W[/tex]

Answer:

Re = 1 10⁴

Explanation:

Reynolds number is

         Re = ρ v D /μ

The units of each term are

       ρ = [kg / m³]

       v = [m / s]

      D = [m]

      μ = [Pa s]

The pressure

      Pa = [N / m²] = [Kg m / s²] 1 / [m²] = [kg / m s²]

      μ = [Pa s] = [kg / m s²] [s] = [kg / m s]

We substitute the units in the equation

      Re = [kg / m³] [m / s] [m] / [kg / m s]

      Re = [kg / m s] / [m s / kg]

      RE = [ ]

Reynolds number is a scalar

Let's evaluate for the given point

Where the data for methane are:

viscosity       μ = 11.2 10⁻⁶ Pa s

the density  ρ = 0.656 kg / m³

       D = 2 in (2.54 10⁻² m / 1 in) = 5.08 10⁻² m

       Re = 0.656 4 2 5.08 10⁻² /11.2 10⁻⁶

       Re = 1.19 10⁴

The Reynolds number is a key parameter in fluid mechanics that signals laminar or turbulent flow. It is a dimensionless quantity determined by fluid properties and flow characteristics.

The Reynolds number, a dimensionless parameter, indicates whether flow is laminar or turbulent in fluid mechanics. It is defined as Re = rho VD/mu, where rho is the fluid density, V is the velocity, D is the diameter, and mu is the fluid viscosity. The value of Reynolds number for methane flowing at 4 m/s through a 2-in-diameter pipe is calculated using this formula.

Answer:

.c. −160°C

Explanation:

In the whole process one kg of water at  0°C loses heat to form one kg of ice and heat lost by them is taken up by ice at −160°C . Now see whether heat lost is equal to heat gained or not.

heat lost by 1 kg of water at  0°C

= mass x latent heat

= 1 x 80000 cals

= 80000 cals

heat gained by ice at −160°C to form ice at  0°C

= mass x specific heat of ice x rise in temperature

= 1 x .5 x 1000 x 160

= 80000 cals

so , heat lost = heat gained.

The original temperature of the ice was a. one or two degrees below 0°C.

Heat transfer is the process by which thermal energy is exchanged between systems. It occurs through conduction, where heat moves through materials, convection, involving the movement of fluids, and radiation, which involves electromagnetic waves. Understanding heat transfer is essential in fields like physics, engineering, and environmental science.

To find the original temperature of the ice, we need to calculate the heat transferred during the process. First, we need to bring the ice up to 0°C and melt it. This requires a heat transfer of 4.74 kcal. This will lower the temperature of the water by 23.15°C. After the ice has melted and the system reaches equilibrium, the final temperature of the water is 20.6°C. Therefore, the original temperature of the ice was one or two degrees below 0°C.

Utilizing the principles of impulse and momentum, the magnitude of the impulsive force exerted by the steel plate on the bullet is approximately 40,576.47 N, and its direction is opposite to the bullet's initial motion.

Impulsive Force on a Ricocheting Bullet

The problem involves finding the impulsive force exerted by the steel plate on a steel-jacketed bullet that ricochets off its surface. To determine the impulsive force, we utilize the concepts of impulse and momentum. The bullet has an initial velocity of 630 m/s and exits with a velocity of 500 m/s after ricocheting. While in contact with the plate, we are given an average speed of 600 m/s. The displacement during contact is 50 mm, equivalent to 0.050 meters.

To find the time of contact, we use the formula for distance, which is: distance = average speed * time, or time = distance / average speed. Hence, the time of contact is 0.050 meters / 600 m/s = 0.0000833 seconds (or 83.3 microseconds). Now, using the change in momentum (Δp = mass * change in velocity) and the impulse-momentum theorem (impulse = Δp), we can find the impulse. Considering the magnitude of the velocities and the mass of the bullet (26 g or 0.026 kg), the change in velocity is (500 m/s - 630 m/s) = -130 m/s. Therefore, Δp = 0.026 kg * -130 m/s = -3.38 kg*m/s. Given that impulse equals the average force times the time of the impact (Impulse = average force * time), we can find the average force as Impulse/time = -3.38 kg*m/s / 0.0000833 s = -40,576.47 N. The negative sign indicates that the force on the bullet is directed opposite to its initial motion, which is consistent with a ricochet.

The magnitude of the impulsive force exerted by the plate on the bullet is approximately 40,576.47 N, and its direction is opposite to the bullet's initial motion, demonstrating the principle of conservation of momentum during collisions.

Final answer:

We can estimate the power required to tow a banner by using the drag force formula. If the airplane was towing a rigid flat plate, the power required would be different due to a different drag coefficient. The power requirement and drag are larger for the banner due to its higher drag coefficient compared to a flat plate. No calculation can be made for a smooth spherical balloon without the drag coefficient.

Explanation:

(a) To estimate the power required to tow the banner, we can use the formula: Power = Drag force * velocity. The drag force can be calculated using the drag coefficient and the area of the banner. The area is given by multiplying the height (0.8 m) by the length (25 m). The drag force can then be multiplied by the velocity of the plane (150 km/hr converted to m/s) to obtain the power required.

(b) If the plane was towing a rigid flat plate of the same size, the drag coefficient would be different. The power required can be calculated using the new drag coefficient and the same formula as in part (a).

(c) The power requirement and drag are larger for the banner because the drag coefficient for the banner is higher compared to that of a rigid flat plate. This means that the banner experiences more air resistance, requiring more power to tow.

(d) To determine the power required to tow a smooth spherical balloon with a diameter of 2 m, we would need the drag coefficient associated with the balloon. Since it is not provided in the question, we cannot calculate the power required.

The energy change in an electron's transition in a hydrogen atom can be calculated using the Bohr formula. This energy corresponds to the energy of the emitted photon in the transition, and can be used to calculate the wavelength of the light emitted during this transition.

The energy change associated with the transition of an electron between two energy levels in a hydrogen atom can be calculated using the equation E = 13.6 eV/n². In this equation, 'E' represents the energy of the electron, and 'n' is the quantum number of the orbit that the electron occupies. If we consider a transition of an electron from an orbital with n = 10 to an orbital with n = 8, we can calculate the energy change (∆E) using the difference in energies of these two states. This energy change corresponds to the energy of the emitted photon when the electron transition occurs. The energy of the photon can be connected to its wavelength through the equation E = hf, where 'h' is Planck's constant, and 'f' is the frequency, which is related to the wavelength (λ) by the speed of light (c) as f = c/λ. Therefore, you can first calculate ∆E using the given energy equation, then use the result to find 'f' using E = hf, and finally substitute 'f' in f = c/λ to find λ, the wavelength of the emitted light.

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Answer:

(a) -202 m/s²

(b) 198 m

Explanation:

Given data

[tex]\frac{632mi}{h} .\frac{1609.34m}{1mi} .\frac{1h}{3600s} =283m/s[/tex]

(a) The acceleration (a) is the change in the speed over the time elapsed.

a = (vf - v₀)/t = (0 - 283 m/s)/ 1.40s = -202 m/s²

(b) We can find the distance traveled (d) using the following kinematic expression.

y = v₀ × t + 1/2 × a × t²

y = 283 m/s × 1.40 s + 1/2 × (-202 m/s²) × (1.40 s)²

y = 198 m

Final answer:

The conductivity of the Styrofoam material is found using the heat required to melt the ice and the conduction formula, resulting in 2.8 × 10^⁻⁶cal/s·cm·°C.

Explanation:

To find the conductivity of the Styrofoam material, we need to calculate the amount of heat transferred to the ice cube to melt it completely and then relate this to the conductivity formula. First, we calculate the total heat required to melt the 500-g ice cube using the latent heat of fusion (Lf). The total heat (Q) required is the mass (m) of the ice times the latent heat of fusion (Lf).

Q = m × Lf = 500 g × 80 cal/g = 40000 cal

Next, we use the formula for heat conduction, which is Q = (k × A × ΔT × t) / d. Here, Q is the heat transferred, k is the thermal conductivity, A is the area through which heat is transferred, ΔT is the temperature difference between the inside and outside of the box, t is the time, and d is the thickness of the walls.

Since all values except k are known, we rearrange the equation to solve for k:

k = (Q × d) / (A × ΔT × t)

k = (40000 cal × 1.0 cm) / (600 cm² × 20°C × (4 × 3600 s))

Upon calculation, we find the conductivity k equals 2.8 × 10-6cal/s·cm·°C, which corresponds to option (b).

Answer:

The potential difference through which an electron accelerates to produce x rays is [tex]1.24\times 10^4\ volts[/tex].                                                  

Explanation:

It is given that,

Wavelength of the x -rays, [tex]\lambda=0.1\ nm=0.1\times 10^{-9}\ m[/tex]

The energy of the x- rays is given by :

[tex]E=\dfrac{hc}{\lambda}[/tex]

The energy of an electron in terms of potential difference is given by :

[tex]E=eV[/tex]

So,

[tex]\dfrac{hc}{\lambda}=eV[/tex]

V is the potential difference

e is the charge on electron

[tex]V=\dfrac{hc}{e\lambda}[/tex]

[tex]V=\dfrac{6.63\times 10^{-34}\times 3\times 10^8}{1.6\times 10^{-19}\times 0.1\times 10^{-9}}[/tex]

V = 12431.25 volts

or

[tex]V=1.24\times 10^4\ volts[/tex]

So, the potential difference through which an electron accelerates to produce x rays is [tex]1.24\times 10^4\ volts[/tex]. hence, this is the required solution.

Answer:

[tex] 735 mm Hg = 0.967 atm= 97991.940 Pa=97.992Pa[/tex]

Explanation:

Previous concepts

Atmospheric pressure is defined as "the force per unit area exerted against a surface by the weight of the air above that surface".

Torricelli shows that we can calculate the atmosphric pressure with a glass tube inside of a tank and with the height we can find the pressure with the relation

[tex]P_{atm}=\rho_{Hg} g h[/tex]

Solution to the problem

For this case we can use the following conversion factor:

[tex]1 atm = 760 mm Hg[/tex]

And if we convert the 735 mm Hg to atm we got this:

[tex]735 mm Hg * \frac{1atm}{760 mm Hg}=0.967 atm[/tex]

And also we can convert this value to Pa and Kpa since we have this conversion factor:

[tex] 1 atm =101325 Pa[/tex]

And if we apply the conversion we got:

[tex]0.967 atm *\frac{101325 Pa}{1 atm}=97991.940 Pa[/tex]

And that correspond to 97.99 Kpa.

Finally we can express the atmospheric pressure on different units for this case :

[tex] 735 mm Hg = 0.967 atm= 97991.940 Pa=97.992Pa[/tex]

The atmospheric pressure (atm) resulting from the mercury column is 0.967 atm.

The given parameters;

The atmospheric pressure (atm) resulting from the mercury column is calculated as follows;

760 mmHg ------- 1 atm

735 mmHg -------- ?

[tex]= \frac{735 \ \times \ 1 atm}{760} \\\\= 0.967 \ atm[/tex]

Thus, the atmospheric pressure (atm) resulting from the mercury column is 0.967 atm.

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Answer:

The Law of Conservation of Mass-Energy

Explanation:

It states that nothing can be created or destroyed, which is why the cakes mass is the same as the ingredients.

Answer : This is an example of law of conservation of mass.

Explanation :

Law of conservation of mass : It states that mass can neither be created nor destroyed but it can only be transformed from one form to another form.

The balanced chemical reaction always follow the law of conservation of mass.

Balanced chemical reaction : It is defined as the reaction in which the number of atoms of individual elements present on reactant side must be equal to the product side.

For example :

[tex]2H_2+O_2\rightarrow 2H_2O[/tex]

This reaction is a balanced chemical reaction in which number of atoms of hydrogen and oxygen are equal on the both side of the reaction. So, this reaction obey the law of conservation of mass.

As per question, the total mass of the cake was equal to the total mass of the ingredients. That means, they obey the law of conservation of mass.

Hence, this is an example of law of conservation of mass.

Answer:

The distance is 20 m.

Explanation:

Given that,

Mass of two blocks = 4.0 kg

Initial velocity = 35 m/s

Angle = 45°

Time = 2 sec

After, they become untied, center of mass trajectory will not change:

We need to calculate the distance

Using equation of motion

[tex]s=ut-\dfrac{1}{2}gt^2[/tex]

Where, u = initial velocity

g = acceleration due to gravity

t = time

[tex]s=0\times2-\dfrac{1}{2}\times9.8\times2^2[/tex]

[tex]s=-19.6 = -20\ m[/tex]

[tex]|s|=20\ m[/tex]

Hence, The distance is 20 m.

The center of mass of the two-block system is approximately 4.4 meters below the highest point 2.0 seconds later.

First, we need to determine the initial vertical velocity of the two-block system. We can use the initial velocity and the launch angle to find the vertical component of the velocity:

Vertical velocity = initial velocity * sin(angle) = 35 m/s * sin(45°)

Then, we can use the formula for vertical displacement to find how far below the highest point the center of mass is:

Vertical displacement = (initial vertical velocity * time) - (0.5 * acceleration * time^2)

Since the blocks are at their highest point, the vertical displacement is equal to zero. We can rearrange the formula to solve for time:

Time = (initial vertical velocity) / acceleration

Finally, we can substitute the values into the formula for vertical displacement:

Vertical displacement = (initial vertical velocity * time) - (0.5 * acceleration * time^2)

The center of mass of the two-block system is approximately 4.4 meters below the highest point 2.0 seconds later.

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Answer:

a. λ = 647.2 nm

b. I₀  9.36 x 10⁻⁵

Explanation:

Given:

β = 56.0 rad , θ = 3.09 ° , γ = 0.170 mm = 0.170 x 10⁻³ m

a.

The wavelength of the radiation can be find using

β = 2 π / γ * sin θ

λ = [ 2π * γ * sin θ ] / β

λ = [ 2π * 0.107 x 10⁻³m * sin (3.09°) ] / 56.0 rad

λ = 647.14 x 10⁻⁹ m  ⇒  λ = 647.2 nm

b.

The intensity of the central maximum I₀

I = I₀ (4 / β² ) * sin ( β / 2)²

I = I₀ (4 / 56.0²) * [ sin (56.0 /2) ]²

I = I₀  9.36 x 10⁻⁵

To find the wavelength of the radiation and the intensity at a specific point in a single-slit diffraction pattern, we can use formulas and calculations based on the given information.

To find the wavelength of the radiation, we can use the formula for single-slit diffraction:

sin(θ) = mλ/w

Where θ is the angle from the center of the central maximum, m is the order of the bright fringe (m = 1 for the first bright fringe), λ is the wavelength, and w is the width of the slit. Rearranging the formula, we get:

λ = wsin(θ)/m

Plugging in the values, we have:

λ = (0.107 mm)(sin(3.09°))/(1)

Calculating this gives us the wavelength of the radiation.

To find the intensity at the given point, we can use the formula for intensity in single-slit diffraction:

I/I0 = (sin(θ)/θ)2

The given point is at an angular distance of 3.09°, so we can use this formula to calculate the intensity.

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Answer:v=10.84 m/s

Explanation:

Given

mass of Cylinder [tex]m=3 kg[/tex]

height of cylinder [tex]h=6 m[/tex]

It is given that tube is evacuated so we can neglect air resistance so friction provided by the air is zero.

Since Energy cannot be destroyed but can be transformed from one form to another therefore Potential Energy of Cylinder is converted to Kinetic Energy of Cylinder

Potential Energy[tex]=mgh=3\times 9.8\times 6=176.4 J[/tex]

Kinetic Energy [tex]=\frac{1}{2}mv^2=\frac{1}{2}\times 3\times (v)^2[/tex]

[tex]176.4=\frac{1}{2}\times 3\times (v)^2[/tex]

[tex]v=\sqrt{117.6}[/tex]

[tex]v=10.84 m/s[/tex]

Speed of cylinder after the cylinder has fallen 6 m is 11 meter/second.

What information do we have?

Mass of cylinder = 3 kg

Height = 6 m

Using energy conservation theroy

mgh = (1/2)mv²

gh = (1/2)v²

(9.8)(6) = (1/2)v²

Velocity = 11 m/s (Approx.)

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Answer:

(a)spring effective force constant =1568N/cm

(b) Yes

Explanation:

Hooke's law is represented mathematically as, F=ke

where

(a).After the maximum load is exceeded, Hooke's law doesn't apply anymore. from the problem, its stated that a maximum load of 120kg will cause an extension of 0.75cm, we will use this to determine the spring constant.

F=mg, g=9.8[tex]m/s^{2}[/tex]

F= 120*9.8 =1176N

From Hooke's law, k=[tex]\frac{F}{e}[/tex]

k=[tex]\frac{1176}{0.75}[/tex]

k=1568N/cm

(b). The players who stands on the scale causes a 0.48cm extension.

    e= 0.48cm, k= 1568N/cm

F=Ke

F= 1568*0.48

F= 752.64N

To calculate the mass of the player we divide this force by g=9.8[tex]m/s^{2}[/tex]

F=mg

m=F/g

[tex]m= \frac{ 752.64}{9.8} \\\\m=76.8kg[/tex]

since the rugby team is an under 85kg team, this player with 76.8kg mass is eligible

Answer:

[tex]\Delta P=128.843\ Pa[/tex]

Explanation:

given,                                

speed of blood = v₂ = 0.800 m/s

v₁ = 0.630 m/s            

density of blood = 1060 kg/m³          

Atmospheric pressure =  8.89 ✕ 10⁴ N/m²

Using  Bernoulli equation                  

[tex]P_1 - P_2 = \dfrac{1}{2}\rho (v_2^2-v_1^2)[/tex]  

[tex]\Delta P= \dfrac{1}{2}\rho (0.8^2-0.63^2)[/tex]            

[tex]\Delta P= \dfrac{1}{2}\times 1060\times (0.8^2-0.63^2)[/tex]            

[tex]\Delta P=128.843\ Pa[/tex]

the change of pressure is equal to [tex]\Delta P=128.843\ Pa[/tex]

Answer : This reaction occur spontaneously at temperature above in kelvins is, 340 K

Explanation : Given,

[tex]\Delta H[/tex] = 17 KJ/mole = 17000 J/mole

[tex]\Delta S[/tex] = 50 J/mole.K

Gibbs–Helmholtz equation is :

[tex]\Delta G=\Delta H-T\Delta S[/tex]

As per question the reaction is spontaneous that means the value of [tex]\Delta G[/tex] is negative or we can say that the value of [tex]\Delta G[/tex] is less than zero.

[tex]\Delta G<0[/tex]

The above expression will be:

[tex]0>\Delta H-T\Delta S[/tex]

[tex]T\Delta S>\Delta H[/tex]

[tex]T>\frac{\Delta H}{\Delta S}[/tex]

Now put all the given values in this expression, we get :

[tex]T>\frac{17000J/mole}{50J/mole.K}[/tex]

[tex]T>340K[/tex]

Therefore, this reaction occur spontaneously at temperature above in kelvins is, 340 K

Final answer:

The reaction occurs spontaneously above 340 Kelvin, determined by applying the Gibbs free energy formula (ΔG = ΔH - TΔS) and ensuring the values for ΔH and ΔS are in compatible units.

Explanation:

The question requires us to determine above what temperature a reaction occurs spontaneously, given an enthalpy change (ΔH) of 17 kJ/mol and an entropy change (ΔS) of 50 J/(mol·K). To find this, we use the formula for Gibbs free energy change (ΔG = ΔH - TΔS), where a reaction is spontaneous when ΔG < 0.

First, we need to ensure that both values are in the same units, so we convert ΔH from kJ to J: 17 kJ/mol = 17000 J/mol. Then, we solve for T in the equation ΔG < 0, substituting ΔH and ΔS into the equation:

0 > 17000 J/mol - T(50 J/(mol·K)),

implying T > 17000 J/mol / 50 J/(mol·K) = 340 K.

Therefore, the reaction occurs spontaneously above 340 K.

Answer:

200.24 kJ

Explanation:

[tex]m[/tex] = mass of sled = 53.9 kg

[tex]d[/tex] = distance traveled by the sled = 1.62 km = 1620 m

[tex]\mu[/tex] = Coefficient of friction between sled and snow = 0.234

frictional force acting on the sled is given as

[tex]f = \mu mg[/tex]

[tex]F[/tex] = Applied force by the dogs on the sled

Since the sled moves at constant speed, the force equation for the motion of the sled is given as

[tex]F = f \\F = \mu mg[/tex]

[tex]W[/tex] = Work done by the dogs on the sled

Work done by the dogs on the sled is given as

[tex]W = F d\\W = \mu mg d\\W = (0.234) (53.9) (9.8) (1620)\\W = 200237.64 J\\W = 200.24 kJ[/tex]

The work done by the dogs is 200.238 kJ.

The work done by the dog is equal to the work done to move the sled through the distance and the work done against friction.

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

Hence, The work done by the dogs is 200.238 kJ.

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Answer:

impedance Z = 416.66 ohm

voltage across inducance V = 346.99 V

power factor = 0.720

Explanation:

given data

resistor R = 300

voltage amplitude = 500 V

resistor = 216 W

to find out

impedance and amplitude of the voltage and power factor

solution

we apply here average power formula that is

average power = I²×R     ............1

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

so

average power =  ([tex]\frac{Vrms}{Z}[/tex])²×R

Vrms = [tex]\frac{1}{\sqrt{2} }[/tex] × Vmax

Z = V × [tex]\sqrt{\frac{R}{2P} }[/tex]

Z = 500 × [tex]\sqrt{\frac{300}{2*216} }[/tex]

impedance Z = 416.66 ohm

and

we know voltage across inductor is here express as

V = I × X     .............2

so here X will be by inductance

Z² = R² + (X)²  

(X)²  = 416.66² - 300²  

X = 289.15 ohm

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

I = [tex]\frac{500}{416.66}[/tex]

I = 1.20 A

so from equation 2

V = 1.20 × 289.15

voltage across inducance V = 346.99 V

and

average power = Vmax × Imax  ×  cos∅

tan∅ = [tex]\frac{289.15}{300}[/tex]

tan∅ = 43.95°

so power factor is

power factor = cos43.95°

power factor = 0.720

The problem involves the use of Ohm's law, impedance, power, and a power factor in an AC circuit. The impedance of the circuit, the voltage across the inductor, and the power factor can be calculated using given values, the principle of impedance and the relationships among AC voltage, current, resistance and power.

The question involves an AC circuit composed of a resistor and inductor in series connected to an AC voltage source. We have a resistor with a resistance (R) of 300 Ω and a dissipated power (P) of 216W. The voltage amplitude (Vo) of the AC source is 500 V. It is important to remember that in this context, impedance (Z), which has a unit of ohms, represents the total resistance in an AC circuit and can be calculated using Ohm's law for AC circuits.

(a) To find the impedance Z of the circuit, we consider that the power P is given by the relation P = Vo^2/R, substituting for P, and R, we can solve for Vo, which will be sqrt(P*R). Then, the rms voltage (V) is given by Vo/sqrt(2). Our current I would be P/V. Finally, applying Ohm's law, Z=V/I would give us the impedance.

(b) The voltage across the inductor can be found by using Pythagoras' Theorem in the context of an AC circuit, VL = sqrt(Vo^2 - VR^2), where VR is the voltage across the resistor (equal to I* R).

(c) Lastly, the power factor can be found as the cosine of the phase angle θ, which can also be defined as R/Z. We'd first calculate θ = arccos(R/Z), and then find the power factor as cos(θ).

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Answer:

Explanation:

Energy stored in a capacitor

= 1/2 C₁V²

capacity of a capacitor

c = εK A / d

k is dielectric and d is distance between plates .

When  the distance between the plates is halved and then filled with a dielectric (κ = 4.3)

capacity becomes 4.3  x 2 times

New capacity

C₂ = 8.6 C₁

Energy of modified capacitor

1/2 C₂ V²= 1/2 x 8.6 c x V²

Energy becomes

8.6 times.

Energy stored = 8.6 x 10⁻⁴ J

Answer:

 w = 0.808 rad / s

Explanation:

As indicated by the moment of inertia t the angular velocity of the disks we use the concept of conservation of the angular momentum, for this we define the system as formed by the two discs, therefore the torque during the crash is internal and the angular momentum is conserved

Let's write in angular momentum

Initial. Before impact

         L₀ = I₁ w₁ + I₂ w₂

Final. After the rock has stuck

          [tex]L_{f}[/tex] = (I₁ + I₂) w

The two discs are rotating in opposite directions, we consider the rotation of the first positive disc, so the angular velocity of the second is negative

          L₀ =[tex]L_{f}[/tex]

           I₁ w₁ - I₂ w₂ = (I₁ + I₂) w

           w = (I₁ w₁ - I₂ w₂) / (I₁ + I₂)

Let's calculate

          w = (0.47 2.0 - 0.31 1.0) / (0.47+ 0.31)

          w = 0.63 / 0.78

          w = 0.808 rad / s

in the direction of disc rotation 1

Answer:

the minimum amount of pressing force P will be 14.29 N

Explanation:

the friction force Fr will be

Fr = μ*N

where μ= coefficient of static friction , N= force normal to the plane

then N=P (force applied by the person)

from Newton's first law

net force = F = 0

Fr - weight = 0

μ*P - m*g =0

P = m*g/μ = 1.05 kg*9.8 m/s² / 0.720 = 14.29 N

P = 14.29 N

Final answer:

The minimum nonzero thickness of an oil film with index of refraction 1.64, between glass with index 1.43 to achieve no reflection for 578 nm light at normal incidence, is approximately 88 nm. This results from the condition for destructive interference.

Explanation:

When light of wavelength 578 nm hits the film at normal incidence and no light is reflected, it means that the reflected light waves from the top and bottom surfaces of the oil film are exactly out of phase, causing destructive interference.

For destructive interference to occur, the path difference between the two reflected waves must be an odd multiple of half the wavelength in the medium, which is given by the formula:

Path Difference (in the medium) = 2 × n × thickness of the film = [tex](m + \(\frac{1}{2}\)) \times \(\frac{\lambda}{n}\)[/tex], where m = 0, 1, 2, ...

Considering that there is a [tex]\(\frac{\lambda}{2}\)[/tex] phase shift when light reflects off a medium with a lower index of refraction to a higher one, and the film's index of refraction is greater than that of the surrounding glass, we take m = 0 for the smallest non-zero thickness. Thus, the minimum thickness of the oil film is:

Thickness = [tex]\(\frac{\lambda}{4 * n}\)[/tex] = [tex]\(\frac{578 nm}{4 \times 1.64}\)[/tex] = 88.16 nm

The nonzero minimum thickness of the oil film that will satisfy the conditions of no reflected light is therefore approximately 88 nm.

Answer:c

Explanation:

it is given that Adam and Boby starts from same height so their total Energy at top is [tex]T_t[/tex]

[tex](T_{top})_a=[/tex]Potential Energy of Adam

[tex](T_{top})_b=[/tex]Potential Energy of boby

when they fall a height h their speed at bottom will be

[tex]v=\sqrt{2gh}[/tex]

which will be same for both Adam and Boby

Energy at bottom

[tex](T_b)_a=[/tex]Kinetic Energy of Adam

[tex](T_b)_b=[/tex]Kinetic Energy of boby

Since velocity is same therefore kinetic Energy is same for Adam and boby

thus option c is correct