The following conversions occur frequently in physics and are very useful. (a) Use 1 mi = 5280 ft and 1 h = 3600 s to convert 60 mph to units of ft/s. (b) The acceleration of a freely falling object is 32 ft/s2. Use 1 ft = 30.48 cm to express this acceleration in units of m/s2. (c) The density of water is 1.0 g/cm3. Convert this density to units of kg/m3.

In order for the net effect at point A to be a downward force, the tension in cable AC (T) should be equal to the tension in cable AB (9.1 kN). The magnitude of the resulting downward force (R) would be the sum of the tensions in both cables, thus 2 * 9.1 kN = 18.2 kN.

To understand this scenario, it is essential to apply the principles of equilibrium and vector sum in Physics. The tension in the wires can be considered as forces experienced by point A. According to the question, the net effect of these forces should be a downward force, implying that they should negate the opposite upward force.

To find the tension T in cable AC, it's logical to assume that the force due to this tension needs to be equal and opposite to the force exerted by the tension in wire AB, which is 9.1 kN. Therefore, T should also be 9.1 kN for the net effect at point A to be a downward force.

The magnitude R of the downward force can be determined by considering the combined effect of tensions in cables AB and AC. Since point A is in equilibrium, R will be the result of the total upward forces. Hence, R is equal to 2 times the tension in any one cable (as they are equal), which gives us R = 2 * 9.1 kN = 18.2 kN.

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Answer:

11995200 N/m²

Explanation:

Pressure: The is the ratio of force to the surface area in contact. The S.I unit of pressure is N/m².

Generally pressure in fluid can be expressed as

P = ρgh......................... Equation 1

Where P = maximum pressure on the submarine hull, ρ = Density of ocean, h = depth of ocean, g = acceleration due to gravity.

Given: h = 1200 m, ρ = 1020 kg/m³

Constant: g = 9.8 m/s²

Substitute into equation 1

P = 1200(1020)(9.8)

P = 11995200 N/m²

Hence the maximum pressure on the submarine hull = 11995200 N/m²

Final answer:

The maximum pressure on a submarine hull that descends 1200 m in the ocean, with ocean density of 1020 kg/m³, is calculated using the formula P = [tex]P_{o}[/tex] + ρgh. With an atmospheric pressure of 101325 Pa, the final pressure would be 12116485 Pa at that depth.

Explanation:

To calculate the maximum pressure on a submarine hull that can descend 1200 m in the ocean, we use the formula for the pressure exerted by a fluid at a certain depth. The formula is P = [tex]P_{o}[/tex] + ρgh, where P is the pressure at depth, [tex]P_{o}[/tex] is the atmospheric pressure on the surface, ρ (rho) is the density of the fluid, g is the acceleration due to gravity, and h is the depth.

Atmospheric pressure [tex]P_{o}[/tex] is approximately 101325 Pa (or 1 atm). The density of sea water is given as 1020 kg/m³ and the depth is 1200 m. Taking g as 9.8 m/s², the standard acceleration due to gravity, we can substitute the values into the formula:

P = 101325 Pa + (1020 kg/m³)×(9.8 m/s²)×(1200 m)

P = 101325 Pa + 12003360 Pa

P = 12116485 Pa

Therefore, the maximum pressure on the submarine hull at a depth of 1200 m would be 12116485 Pascal (Pa).

The reaction produces -4.95 kJ of heat when 15.0 L of CO at 85°C and 112 kPa reacts with 14.4 L of H2 at 75°C and 744 torr.

The equation of the reaction is;

CO(g) + H2(g) -------> CH2O(g)

The heat of reaction is obtained from;

Enthalpy of products - Enthalpy of reactants = (-116kJ/mol)  - (-110.5 kJ/mol)

= -5.5 kJ/mol

Number of moles of CO is obtained from;

PV = nRT

P =  112 kPa or 1.1 atm

T = 85°C + 273 = 358 K

n = ?

R = 0.082 atmLK-1mol-1

V = 15.0 L

n = PV/RT

= 1.1 atm × 15.0 L/ 0.082 atmLK-1mol-1 ×  358 K

= 0.56 moles

Number of moles of H2

n = PV/RT

P= 744 torr or 0.98 atm

V = 14.4 L

T = 75°C + 273 = 348 K

n =  0.98 atm ×  14.4 L/0.082 atmLK-1mol-1 ×  348 K

n = 0.49 moles

We can see that H2 is the limiting reactant here hence 0.49 moles of formaldehyde is produced.

If 1 mole of formaldehyde produces -5.5 kJ of heat

0.49 moles of formaldehyde produces -5.5 kJ ×  0.49 moles / 1 mole

= -4.95 kJ of heat

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To solve this problem we will apply the concepts related to the relationship between energy and frequency, from the latter we will obtain similar expressions that relate to the wavelength to find the two energy states between the given values. Finally we will make the comparative radius between the two. The relation between energy and frequency is given as,

[tex]E = hf[/tex]

Here,

E = Energy

h = Planck's constant

The relation between the speed of the electromagnetic waves (c), frequency (f) and wavelength ([tex]\lambda[/tex] ) is,

[tex]c = f\lambda[/tex]

Rearrange the above equation for frequency f as follows

[tex]f = \frac{c}{\lambda}[/tex]

Substitute,

[tex]E = h\frac{c}{\lambda}[/tex]

The wavelength x-ray or gamma ray photon is

[tex]\lambda = 1.0nm (\frac{1nm}{10^{9}nm})[/tex]

[tex]\lambda = 10^{-9} m[/tex]

Therefore the energy would be,

[tex]E_1 = \frac{hc}{\lambda}[/tex]

[tex]E_1 = \frac{(6.63*!0^{-34}J\cdo s)(3*10^{8}m/s)}{10^{-9}m}[/tex]

[tex]E_1 = 19.89*10^{-17} J[/tex]

The frequency is given as,

[tex]f = 10MHz (\frac{10^6z}{1.0MHz})[/tex]

[tex]f = 10^7Hz[/tex]

Now the second energy would be

[tex]E_2 = hf[/tex]

[tex]E_2 = (6.63*10^{-27}J\cdot s)(10^7Hz)[/tex]

[tex]E_2 = 6.63*10^{-27}J[/tex]

Therefore the ratio between them is

[tex]\frac{E_1}{E_2} = \frac{19.89*10^{-17}J}{6.63*10^{-27}J}[/tex]

[tex]\frac{E_1}{E_2} = 3*10^{20}[/tex]

Therefore the energy of 1nm x ray or gamma ray photon is [tex]3*10^{20}[/tex] times more than energy of 10MHz radio photon

Answer:

a.Insulators can be used to increase the amount of current that can flow through a resistor without increasing its temperature.

Explanation:

A conductor has low resistance, while an insulator has much higher resistance. Devices called resistors control amounts of resistance into an electrical circuits. Electric charges inside an insulator are bound to the individual atoms or molecules, not being able to move inside the material.

When you turn up the resistance, the electric current flowing through the circuit is reduced

Ohm's law:

V = I * R

R = V/I

An Ohmic conductor would have a linear relationship between the current and the voltage. With non-Ohmic conductors, the relationship is not linear. Most metals are good conductors example silver, copper etc.

Answer:

P = 0.533 W

Explanation:

given,

Resistance of the bulb, R = 270 Ω

Potential of the battery, V=  12 V

Power output of the bulb = ?

we know,

P = I² R

also, V = IR

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

[tex]P =\dfrac{12^2}{270}[/tex]

[tex]P =\dfrac{144}{270}[/tex]

     P = 0.533 W

Hence, the Power delivered by the bulb is equal to 0.533 W.

Answer:

Air Resistance

Explanation:

If you were to drop both items on a plant without an atmosphere, they would both hit the ground at the same time. Since a feather doesn't have much mass compared to the hammer, it takes more time for the feather to "push" itself through and overcome the opposite push from the air

Final answer:

On Earth, air resistance prevents a feather and a hammer from hitting the ground simultaneously when dropped from the same height. On the Moon, the absence of an atmosphere means no air resistance, allowing both objects to land at the same time in accordance with Galileo's principle of the universality of free fall.

Explanation:

If you drop a feather and a steel hammer at the same moment, they should hit the ground at the same time according to Galileo's principle of the universality of free fall. However, this does not occur on Earth due to the presence of air resistance. The feather experiences a significant amount of air resistance because of its shape and light weight, causing it to flutter and fall slower than the hammer.

On the Moon, where there is no atmosphere, there is no air resistance to act on the objects. When Apollo 15 astronaut David Scott conducted the experiment on the Moon, both the hammer and feather fell at the same acceleration and hit the lunar surface simultaneously. This specific demonstration was a perfect illustration of the universality of free fall in the absence of external forces besides gravity. On the Moon, the acceleration due to gravity is only 1.67 m/s², which is less than on Earth, but since it acts equally on all objects, both the feather and the hammer fell at the same rate.

Answer:

1/2 Hz

Explanation:

A simple harmonic motion has an equation in the form of

[tex]x(t) = Acos(\omega t - \phi)[/tex]

where A is the amplitude, [tex]\omega = 2\pi f[/tex] is the angular frequency and [tex]\phi[/tex] is the initial phase.

Since our body has an equation of  x = 5cos(π t + π/3) we can equate [tex]\omega = \pi[/tex] and solve for frequency f

[tex]2\pi f = \pi[/tex]

f = 1/2 Hz

0.5Hz

The general equation of the displacement, x, of a body undergoing simple harmonic motion at a given point in time (t) is given by;

x = A cos (ωt ± ∅)  --------------------------(i)

where;

A = amplitude of the wave

ω = angular velocity of the wave

∅ = phase constant of the wave

From the question;

x = 5cos(π t + π/3)      -----------------------------(ii)

Comparing equations (i) and (ii), the following deductions among others can be made;

A = 5cm

ω = π

But the angular velocity (ω) of the wave is related to its frequency (f) as follows;

ω = 2 π f        --------------------(iii)

Substitute the value of ω = π  into equation (iii) as follows;

π = 2 π f

Divide through by π;

1 = 2f

Solve for f;

f = 1/2

f = 0.5

Frequency (f) is measured in Hz. Therefore, the frequency of the oscillation is 0.5Hz

Answer:

a) The force of attraction between these two ions at their equilibrium interionic separation (i.e., when the ions just touch one another) is - 2.01 × 10⁻⁹ N

b) The force of repulsion at this same separation distance is 2.01 × 10⁻⁹ N

Explanation:

F = kq₁q₂/r²

r = 0.35 + 0.129 (since the ions are just touching each other)

r = 0.479 nm = 4.79 × 10⁻¹⁰ m

Since the first ion is a divalent cation, Z₁ = +2 and the monovalent anion, Z₂ = -1

q = Ze; e = 1.602 × 10⁻¹⁹ C

K = 8.99 × 10⁹ Nm²/C²

F = (8.99 × 10⁹)(1.602 × 10⁻¹⁹)²(2)(-1)/(4.79 × 10⁻¹⁰)² = - 2.01 × 10⁻⁹ N

b) At equilibrium,

Force of attraction + Force of repulsion = 0

Force of repulsion = -(Force of attraction) = 2.01 × 10⁻⁹ N

Answer:

d. The length of the string is equal to one-half of a wavelength

Explanation:

A stretched string of length L, fixed at both ends, is vibrating in its third harmonic. How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration

a. The length of the sting is equal to one-quarter of a wavelength.b. The length of the string is equal to the wavelength.c. The length of the string is equal to twice the wavelength.d. The length of the string is equal to one-half of a wavelength

e. The length of the string is equal to four times the wavelength

A stretched string of length L fixed at both ends is vibrating in its third harmonic H

How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration

d. The length of the string is equal to one-half of a wavelength

There are two points during vibration , the node and the antinode

the node is the point where the amplitude is zero.

from the third harmonics, there are two nodes. The first node is half of the wavelength which is the closest to the fixed point.

for third harmonics=3/2lamda

Answer:

T= 89.25 K

Explanation:

Given that

V= 590 mph

We know that

1 mph  = 0.44 m/s

That is why ,V= 263.75 m/s

We know that speed of the gas molecule is given as

[tex]V=\sqrt{\dfrac{3RT}{M}}[/tex]

R= 8.314 J/mol.k

M= 32 g/mol = 0.032 kg/mol

T=Temperature in Kelvin unit

[tex]V^2=\dfrac{3RT}{M}[/tex]

[tex]T=\dfrac{V^2\times M}{3R}[/tex]

[tex]T=\dfrac{263.76^2\times 0.032}{3\times 8.314}\ K[/tex]

T= 89.25 K

Therefore the average temperature ,T = 89.25 K

Answer:

temperature does the average speed of an oxygen molecule equal that of an airplane moving at 590 mph  = 89.24 K

Explanation:

Average speed of oxygen molecule is given by

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

[tex]590\times0.44704 = 263.75[/tex] m/s

R= 8.314 J/mol K = universal gas constant

M= molecular weight of oxygen = 32 g/mol =0.032 Kg/mol

now plugging these values to find T we get

[tex]263.75=\sqrt{\frac{3(8.314)(T)}{0.032}}[/tex]

solving the above equation we get

T= 89.24 K

Answer:

heat absorbed = 4.9 × [tex]10^{5}[/tex] J

Explanation:

given data

density of liquid oxygen =  1.14 kg/L

volume = 2 L

heat of vaporization  = 213 kJ/kg

solution

first we get here mass of liquid that is

mass of liquid = density × volume   ......................1

mass of liquid =  1.14 × 2  

mass of liquid = 2.28 kg

so here we get now heat absorbed that is

heat absorbed = mass × heat of vaporization

heat absorbed = 2.28 kg × 213 kJ/kg

heat absorbed = 485.640 kJ

heat absorbed = 4.9 × [tex]10^{5}[/tex] J

To find the energy absorbed by 2.0 L of liquid oxygen as it vaporizes, you first convert the volume to mass using the given density and then multiply by the heat of vaporization. The resulting energy is approximately 4.86 x 10^5 J or 486 kJ.

The energy absorbed by a volume of liquid oxygen as it vaporizes can be calculated using the formula Q = mLv. In this equation, 'm' represents mass, 'Lv' represents the heat of vaporization, and 'Q' represents the total energy absorbed.

Firstly, convert the volume of liquid oxygen to mass. The density of liquid oxygen at its boiling point is 1.14 kg/L, so the mass of 2.0 L of liquid oxygen would be (1.14 kg/L) * (2.0 L) = 2.28 kg.

Then, use the heat of vaporization and the calculated mass to find the total energy. The heat of vaporization of oxygen is 213 kJ/kg, so Q = (2.28 kg) * (213 kJ/kg) = 485.64 kJ. This needs to be expressed in joules by multiplying by 10^3, resulting in 485640 J. Therefore, the energy absorbed by 2.0 L of liquid oxygen as it vaporizes is approximately 4.86 x 10^5 J.

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Answer:

[tex]\omega = \sqrt{\dfrac{kq_0Q}{ma^3} }[/tex]

Explanation:

Additional information:

The ball has charge [tex]-q_0[/tex], and the ring has  positive charge [tex]+Q[/tex] distributed uniformly along its circumference.

The electric field at distance [tex]z[/tex] along the z-axis due to the charged ring is

[tex]E_z= \dfrac{kQz}{(z^2+a^2)^{3/2}}.[/tex]

Therefore, the force on the ball with charge [tex]-q_0[/tex] is

[tex]F=-q_oE_z[/tex]

[tex]F=- \dfrac{kq_0Qz}{(z^2+a^2)^{3/2}}[/tex]

and according to Newton's second law

[tex]F=ma=m\dfrac{d^2z}{dz^2}[/tex]

substituting [tex]F[/tex] we get:

[tex]- \dfrac{kq_0Qz}{(z^2+a^2)^{3/2}}=m\dfrac{d^2z}{dz^2}[/tex]

rearranging we get:

[tex]m\dfrac{d^2z}{dz^2}+ \dfrac{kq_0Qz}{(z^2+a^2)^{3/2}}=0[/tex]

Now we use the approximation that

[tex]z^2+a^2\approx a^2[/tex] (we use this approximation instead of the original [tex]d^2+a^2\approx a^2[/tex] since [tex]z<d[/tex], our assumption still holds )

and get

[tex]m\dfrac{d^2z}{dz^2}+ \dfrac{kq_0Qz}{(a^2)^{3/2}}=0[/tex]

[tex]m\dfrac{d^2z}{dz^2}+ \dfrac{kq_0Qz}{a^{3}}=0[/tex]

Now the last equation looks like a Simple Harmonic Equation

[tex]m\dfrac{d^2z}{dz^2}+kz=0[/tex]

where

[tex]\omega=\sqrt{ \dfrac{k}{m} }[/tex]

is the frequency of oscillation. Applying this to our equation we get:

[tex]m\dfrac{d^2z}{dz^2}+ \dfrac{kq_0Q}{a^{3}}z=0\\\\m=m\\\\k= \dfrac{kq_0Q}{a^{3}}[/tex]

[tex]\boxed{\omega = \sqrt{\dfrac{kq_0Q}{ma^3} }}[/tex]

Answer:

Explanation:

Given

Electric Field Strength [tex]E=4.5\times 10^{3}\ V/m[/tex]

Potential Difference between Plates is given by [tex]V=12.5\ kV[/tex]

In conducting plates a Potential difference exist between two plate which accelerate the charge when put between the conducting plates

The potential difference is given by

[tex]\Delta V=Ed[/tex]

where E=Electric Field strength

d=distance between Plates

[tex]d=\frac{\Delta V}{E}[/tex]

[tex]d=\frac{12.5\times 10^3}{4.5\times 10^{3}}[/tex]

[tex]d=2.78\ m[/tex]

Answer:

Explanation:

Given

speed of Electron [tex]u=2\times 10^7\ m/s[/tex]

final speed of Electron [tex]v=4\times 10^7\ m/s[/tex]

distance traveled [tex]d=1.2\ cm[/tex]

using equation of motion

[tex]v^2-u^2=2as[/tex]

where v=Final velocity

u=initial velocity

a=acceleration

s=displacement

[tex](4\times 10^7)^2-(2\times 10^7)^2=2\times a\times 1.2\times 10^{-2}[/tex]

[tex]a=5\times 10^{16}\ m/s^2[/tex]

acceleration is given by [tex]a=\frac{qE}{m}[/tex]

where q=charge of electron

m=mass of electron

E=electric Field strength

[tex]5\times 10^{16}=\frac{1.6\times 10^{-19}\cdot E}{9.1\times 10^{-31}}[/tex]

[tex]E=248.3\ kN/C[/tex]                

Answer:

1.2 cm

Explanation:

Thermal Expansion

It's the tendency that materials have to change its size and/or shape under changes of temperature. It can be in one (linear), two (surface) or three (volume) dimensions.

The formula to compute the expansion of a material under a change of temperature from [tex]T_o[/tex] to [tex]T_f[/tex] is given by.

[tex]\Delta L=L_o.\alpha .(T_f-T_o)[/tex]

Where Lo is the initial length and [tex]\alpha[/tex] is the linear temperature expansion coefficient, which value is specific for each material. The data provided in the problem is as follows:

[tex]L_o=30\ m,\ T_f-T_o=30^oC,\ \alpha=13\times 10^{-6}\ ^oC^{-1}[/tex]

Computing the expansion we have

[tex]\Delta L=30\times 13\times 10^{-6}(30)=0.0117=1.17\ cm[/tex]

The expansion gap should be approximately 1.2 cm

To solve this problem it is necessary to apply the concepts related to density, such as the relationship between density and Volume.

The volume of a sphere can be expressed as

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

Here r is the radius of the sphere and V is the volume of Sphere

Using the expression of the density we know that

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

The density is given as

[tex]\rho = (19.5g/cm^3)(\frac{10^3kg/m^3}{1g/cm^3})[/tex]

[tex]\rho = 19.5*10^3kg/m^3[/tex]

Now replacing the mass given and the actual density we have that the volume is

[tex]V = \frac{60kg}{19.5*10^3kg/m^3 }[/tex]

[tex]V = 3.0769*10^{-3} m ^3[/tex]

The radius then is,

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

[tex]r = \sqrt[3]{\frac{3V}{4\pi}}[/tex]

Replacing,

[tex]r = \sqrt[3]{\frac{3(3.0769*10^{-3})}{4\pi}}[/tex]

The radius of a sphere made of this material that has a critical mass is 9.02 cm.

Answer:

[tex]T=94.54N[/tex]

Explanation:

The tension in a cable is given by:

[tex]T=\mu v^2(1)[/tex]

Where [tex]\mu[/tex] is the mass density of the cable and v is the speed of the cable's pulse. These values are defined as:

[tex]\mu=\frac{m}{L}(2)\\v=\frac{d}{t}[/tex]

The pulse makes four trips down and back along the cable, so [tex]d=4(2L)[/tex]

[tex]v=\frac{8L}{t}(3)[/tex]

Replacing (2) and (3) in (1), we calculate the tension in the cable:

[tex]T=\frac{m}{L}(\frac{8L}{t})^2\\T=\frac{64mL}{t^2}\\T=\frac{64(0.21kg(3.80m))}{(0.735s)^2}\\T=94.54N[/tex]

Answer:

Electric potential, [tex]V=-4.01\times 10^4\ volts[/tex]

Explanation:

Given that,

Charge 1, [tex]q_1=2.2\ \mu C[/tex]

Point charge 2, [tex]q_2=-8\ \mu C[/tex]

Distance between charges, d = 2.6 m

We need to find the electric potential midway between them. The electric potential is given by :

[tex]V=\dfrac{kq}{r}[/tex]

In this case, r = 1.3 m (midway between charges)

[tex]V=\dfrac{kq_1}{r}-\dfrac{kq_2}{r}[/tex]

[tex]V=\dfrac{k}{r}(q_1-q_2)[/tex]

[tex]V=\dfrac{9\times 10^9}{1.3}(2.2\times 10^{-6}-8\times 10^{-6})[/tex]

[tex]V=-40153.84\ volts[/tex]

or

[tex]V=-4.01\times 10^4\ volts[/tex]

So, the electric potential midway between the charges is [tex]V=-4.01\times 10^4\ volts[/tex]. Hence, this is the required solution.

Final answer:

The electric potential midway between two point charges of +2.20 μC and -8.00 μC, separated by 2.60 m, is calculated separately for each charge using the formula V = kq/r and summed up. The total electric potential at the midpoint is -4.00 × 10⁴ V.

Explanation:

To find the electric potential midway between two point charges, we need to consider the contribution from each charge separately and then sum them up.

The electric potential V due to a single point charge q at a distance r is given by the formula:

V = kq/r

where k is the Coulomb's constant (k ≈ 8.99 × 109 Nm²/C²).

In this case, we have two charges, +2.20 μC and -8.00 μC, and they are separated by 2.60 m. So the distance from the midpoint to each charge is 1.30 m (half of 2.60 m).

Calculating the potential due to the +2.20 μC charge:

V₁ = (8.99 × 109)(+2.20 × 10⁻⁶) / 1.30 = 1.53 × 10⁴ V

And for the -8.00 μC charge:

V₂ = (8.99 × 10⁹)(-8.00 × 10⁻⁶) / 1.30 = -5.53 × 10⁴ V

The total electric potential at the midpoint is the sum of V₁ and V₂:

Vtotal = V₁ + V₂ = 1.53 × 10⁴ V - 5.53 × 10⁴ V = -4.00 × 10⁴V

The electric potential midway between the two charges is -4.00 × 10⁴V.

Explanation:

The given data is as follows.

       Mass of evaporating dish = 3.375 g

    Total mass = Mass of solid sample + evaporating dish

That is, Mass of solid sample + evaporating dish = 26.719 g

Therefore, we will calculate the mass of solid sample as follows.

    Mass of solid sample = (Mass of solid sample + evaporating dish) - mass of evaporating dish

                            = 26.719 g – 3.375 g

                            = 23.344 g

Thus, we can conclude that mass of his solid sample must be 23.344 g.

The mass of the solid sample is 23.344 g.

In order to find the mass of the solid sample, we need to subtract the mass of the evaporating dish from the combined mass of the dish and the sample. The mass of the solid sample can be calculated by subtracting 3.375 g (mass of the evaporating dish) from 26.719 g (combined mass of dish and sample).

Mass of solid sample = 26.719 g - 3.375 g = 23.344 g.

Therefore, the mass of the solid sample is 23.344 g.

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Answer:

Explanation:

Let the vertical height by which it descends be h . Let it acquire velocity of v .

1/2 mv² = mgh

v² = 2gh

As it leaves the surface of sphere , reaction force of surface  R = 0 , so

centripetal force = mg cosθ where θ is the angular displacement from the vertex .  

mv² / r = mg cosθ

(m/r )x 2gh = mg cosθ

2h / r = cosθ

cosθ = (r-h) / r

2h / r =  r-h / r

2h = r-h

3h = r

h = r / 3

Final answer:

Through conservation of energy and dynamics principles, the puck descends through a height of R/2 from the base of the sphere before losing contact, due to the gravitational force no longer providing sufficient centripetal force.

Explanation:

The question asks through what vertical height a small frictionless puck will descend before it leaves the surface of a fixed sphere of radius R, when given a tiny nudge down the sphere. Using the principles of energy conservation and dynamics, it can be determined that the puck will lose contact with the sphere when the centripetal force is no longer sufficient to provide the necessary force for circular motion, which happens at a height of R/2 from the base of the sphere. This happens because, at this point, the gravitational component acting towards the center of the sphere is exactly equal to the required centripetal force for circular motion. As a result, any further descent would mean this balance is disturbed, causing the puck to leave the surface of the sphere.

Answer:

[tex]x=9.78\times 10^0[/tex]

Explanation:

In this case, we need to find the value of expression :

[tex]x=\dfrac{(6.67\times 10^{-11})\times (5.97\times 10^{24})}{(6.38\times 10^6)^2}[/tex]

On solving, we get the value of given expression as :

x = 9.7827

In scientific notation, we get the value of x as :

[tex]x=a\times 10^k[/tex]

[tex]x=9.78\times 10^0[/tex]

a = 9.78

k = 0

Hence, this is the required solution.

To express the given expression in scientific notation, we find a = 9.78 and k = -1 after evaluating the expression using the laws of exponentiation and division.

The student's question involves evaluating an expression using scientific notation and expressing the result in proper scientific notation.

To solve (6.67×10−11)(5.97×1024) / (6.38×106)2, we need to use the laws of exponentiation and multiplication. Simplify the expression within the numerator and denominator separately before dividing them.

Numerator: (6.67×10−11)(5.97×1024) = 39.8309×1013

Denominator: (6.38×106)2 = 40.7044×1012

Divide the simplified numerator by the simplified denominator:
39.8309×1013 / 40.7044×1012 = 0.978×101

To express this in proper scientific notation, rewrite 0.978×101 as 9.78×10−0.

Therefore, the values of a and k when the value of this expression is written in scientific notation are a = 9.78 and k = −0.

Answer:

a. [tex]t=3.44s[/tex]

b. [tex]x=12.45m[/tex]

c. [tex]v_f=7.26\frac{m}{s}[/tex]

Explanation:

The bicyclist's friend moves with constant speed. So, we have:

[tex]x=vt[/tex]

Th bicyclist moves with constant acceleration and starts at rest ([tex]v_0=0[/tex]). So, we have:

[tex]x=v_0t+\frac{at^2}{2}\\x=\frac{at^2}{2}[/tex]

a. When he catches his friend, both travels the same distance, thus:

[tex]vt=\frac{at^2}{2}\\t=\frac{2v}{a}\\t=\frac{2(3.63\frac{m}{s})}{2.11\frac{m}{s^2}}\\t=3.44s[/tex]

b. We can use any of the distance equations, since both travels the same distance:

[tex]x=vt\\x=3.63\frac{m}{s}(3.44s)\\x=12.45m[/tex]

c. The bicyclist final speed is:

[tex]v_f=v_0+at\\v_f=at\\v_f=2.11\frac{m}{s^2}(3.44s)\\v_f=7.26\frac{m}{s}[/tex]

Answer:

[tex]-43\hat{k}[/tex]

Explanation:

given,

[tex]\vec{A} = 4 \hat{i} + 7 \hat{j}[/tex]

[tex]\vec{B} = 5 \hat{i} - 2 \hat{j}[/tex]

vector product [tex] \vec{A} \times \vec{B} = ?[/tex]

[tex]\vec{A} \times \vec{B}[/tex] = [tex]\begin{bmatrix}i & j & k\\ 4 & 7 &0 \\ 5 & -2 & 0\end{bmatrix}[/tex]

now, expanding the vector

[tex]\vec{A} \times \vec{B}= \hat{k}(-2\times 4 - 7\times 5)[/tex]

[tex]\vec{A} \times \vec{B}= -43\hat{k}[/tex]

the vector product is equal to [tex]-43\hat{k}[/tex]

Answer:

[tex]F=0.26N[/tex]

Explanation:

Assuming that thet act like point charges, the attractive force is given by Coulomb's law:

[tex]F=\frac{kq_1q_2}{d^2}[/tex]

Where k is the Coulomb constant, [tex]q_1[/tex] and [tex]q_2[/tex] are the magnitudes of the point charges and d is the distance of separation between them. Thus, we replace the given values and get how strong is the attractive force between them:

[tex]F=\frac{8.99*10^{9}\frac{N\cdot m^2}{C^2}(0.7*10^{-6}C)(-0.6*10^{-6}C)}{(12*10^{-2}m)^2}\\F=0.26N[/tex]

Answer:

E = 7.77 N/C

Explanation:

The charge of a single electron is 1.6 x 10^{-19} C. The net charge of the object is therefore the multiplication of the number of excess electrons and the charge of a single electron:

[tex]Q = (2.6\times 10^4) \times 1.6\times 10^{-19} = 4.16 \times 10^{-15}~C[/tex]

The electric field can be found by the following formula

[tex]E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}[/tex]

where 'r' can be calculated as

[tex]r = \sqrt{(2\times 10^{-3})^2 + (1\times 10^{-3})^2} = 0.0022~m\\r^2 = 4.84\times 10^{-6}[/tex]

Finally, the electric field at the position (2.00 mm, 1.00 mm) is

[tex]E = \frac{1}{4\pi(8.8\times 10^{-12})}\frac{4.16\times 10^{-15}}{4.84\times 10^{-6}} = 7.77~N/C[/tex]

The magnitude of the electric field it produces at the position is 7.5 N/C.

The given parameters:

The position of the charged object is calculated as follows;

[tex]r^2 = (2.0 \times 10^{-3})^2 + (1.0 \times 10^{-3})^2\\\\r^2 = 5\times 10^{-6} \ m^2[/tex]

The charge of the electron is calculated as follows;

[tex]Q = nq\\\\Q = 2.6 \times 10^4 \times 1.6\times 10^{-19}\\\\Q =4.16 \times 10^{-15} \ C[/tex]

The magnitude of the electric field it produces at the position is calculated as follows;

[tex]E = \frac{F}{Q}= \frac{kQ}{r^2} = \frac{9\times 10^9 \times 4.16 \times 10^{-15}}{5\times 10^{-6}} \\\\E = 7.5 \ N/C[/tex]

Learn more about electric field here: https://brainly.com/question/4440057

Answer:

Power, [tex]P=3.93\times 10^{26}\ W[/tex]

Explanation:

Given that,

The distance from the earth to the sun is about, [tex]d=1.5\times 10^{11}\ m[/tex]

let us assume that the average intensity of solar radiation at the upper atmosphere of earth,. [tex]I=1390\ W/m^2[/tex]

We need to find the total power radiated by the sun. The intensity is defined as the total power divided by the area of a sphere of radius equal to the average distance between the earth and the sun. It is given by :

[tex]I=\dfrac{P}{4\pi r^2}[/tex]

P is total power

[tex]P=4\pi r^2\times I[/tex]

[tex]P=3.93\times 10^{26}\ W[/tex]

So, the total power radiated by the sun is [tex]P=3.93\times 10^{26}\ W[/tex]. Hence, this is the required solution.

Final answer:

The total power radiated by the Sun is calculated using the area of the Sun and the power radiated per square meter at its surface. The Sun's total power output is found to be 3.82×1026 W. This value is important for understanding the energy Earth receives from the Sun, known as the solar constant (1360 W/m²).

Explanation:

The distance from the Earth to the Sun is about 1.50×1011 meters. The total power radiated by the Sun can be determined by using the following physics principle: The Sun, behaving as a perfect black body with an emissivity of exactly 1, radiates power uniformly across its surface area. The power radiated per square meter on the Sun's surface is found to be 6.3×107 W/m². The sun's radius is approximately 7.00×108 meters. Using the formula Power = Area × Intensity, we calculate the Sun's total power output.

The area of a sphere is given by 4πR2, where R is the radius of the sphere. Therefore, the total power output of the Sun is 4πR2σT4, which calculates to 3.82×1026 W. This immense amount of power is what we refer to as the solar luminosity.

At the distance of the Earth, this power is spread over a spherical area with a radius equal to the Earth-Sun distance. We can find the power per square meter at this distance, known as the solar constant, which is approximately 1360 W/m².

Answer:

a) They will hit the ground with a speed of 19.6 m/s.

b) They are at a height of 20 m.

c) It is not a safe jump.

Explanation:

Hi there!

a) The equations of height and velocity in function of time of a free falling body are the following:

h = h0 + v0 · t + 1/2 · g · t²

v = v0 + g · t

Where:

h = height of the object at time t.

h0 = initial height.

v0 = initial velocity.

t = time.

g = acceleration due to gravity (-9.8 m/s² considering downward as negative direction).

v = velocity of the object at time t.

Using the equation of velocity, let's find the velocity at which they will hit the ground. The pebble is dropped (initial velocity = 0) and it takes 2 s to reach the ground:

v = v0 + g · t     (v0 = 0)

v = g · t

v = -9.8 m/s² · 2.0 s

v = -19.6 m/s

They will hit the ground with a speed of 19.6 m/s.

b)Now, we have to use the equation of height:

h = h0 + v0 · t + 1/2 · g · t²

If we place the origin of the frame of reference on the ground, we have to find the initial height (h0) knowing that at t = 2.0 s, h = 0 m

0 m = h0 - 1/2 · 9.8 m/s² · (2.0 s)²

h0 = 1/2 · 9.8 m/s² · (2.0 s)²

h0 = 20 m

They are at a height of 20 m.

c)According to a NASA paper (Issues on Human Acceleration Tolerance After Long-Duration Space Flights, figure 10), if you fall with a vertical velocity greater than 17 m/s it is unlikely that you will survive. So, it is not a safe jump.  

Answer:

Explanation:

Given

Load [tex]W=20000\ N[/tex]

height to which load is raised [tex]h=250\ m[/tex]

Another crane take [tex]\frac{1}{6}[/tex] th time to lift the load

Energy required required to lift the Weight

[tex]E=W\times h[/tex]

[tex]E=20000\times 250[/tex]

[tex]E=5,000,000\ J[/tex]

Suppose [tex]P_1[/tex] and [tex]P_2[/tex] is the Power required to lift the weight in t and [tex]\frac{t}{6}[/tex] time

[tex]E=P_1\times t[/tex]

[tex]E=P_2\times \frac{t}{6}[/tex]

[tex]P_1\times t=P_2\times \frac{t}{6}[/tex]

thus

[tex]P_2=6P_1[/tex]

Second Crane requires 6 times  more power than the slow crane                                              

The power of the faster crane is 6 times greater than the power of the slower crane.

To calculate the power of the cranes, we need to use the formula:

Power = Work/Time

The work done by each crane is equal to the maximum load lifted multiplied by the height lifted, so:

Work = Load x Height

Let's assume the time taken by the slower crane is t. Therefore, the time taken by the faster crane is t/6.

Now, let's calculate the power of each crane:

Power of slower crane = Work/Time = (20000 N x 250 m) / t = 5000000 Nm/t

Power of faster crane = Work/Time = (20000 N x 250 m) / (t/6) = 30000000 Nm/t

The power of the faster crane is 6 times greater than the power of the slower crane.

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