Now the same particle is removed from the thread and placed over the center of a charged plate. Are there any conditions under which it is possible for the particle to be suspended in the air above the plate? Show any relevant calculations and explain your reasoning.

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

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. [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]

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:

[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:

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:

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:

B. It is an inelastic collision because the train cars stick together and move as one.

Explanation:

Momentum

When two or more objects collide in a closed system (no external forces are acting) the total momentum is conserved:

[tex]m_1v_1+m_2v_2+...+m_nv_n=m_1v_1'+m_2v_2'+...+m_nv_n'[/tex]

where m1...m2 are the masses of each object, v1...vn are their velocities before the collision takes place and v'1...v'n are their velocities after the collision.

If a collision is elastic, then the kinetic energy is also conserved. when the collision is inelastic, part of the initial kinetic energy is lost. A typical case of inelastic collision occurs when the objects join and remain together after the collision. The velocity is common to all of them and the mass is the sum of the individual masses.

This is exactly the case described in the question: serveral train cars are joined and continue moving together after the collision. It corresponds to a inelastic collision described in the option B.

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) 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

This is a physics problem related to Boyle's Law, which says that the volume of a gas decreases if its pressure increases, assuming constant temperature. We can't solve for the final volume specifically in this case since the initial pressure and volume aren't given, but if they were, we'd use the formula V₂ = (P₁V₁) / P₂, where P₁ and V₁ are the initial pressure and volume, and P₂ and V₂ are the final pressure and volume.

This question is related to the concept of Boyle's Law in physics, which states that the pressure and volume of a gas have an inverse relationship, when temperature is held constant. If the pressure of the gas inside a cylinder is increased, the volume decreases, assuming the amount of gas and the temperature remain constant.

In this case without knowing the initial pressure and volume of the cylinder, it's impossible to calculate the exact final volume after the pressure has increased to 1.6 atm. However, the base formula you'd use to find out your final volume, given you had initial values for volume (V₁) and pressure (P₁), is: P₁V₁ = P₂V₂.

In this formula, P₁ and V₁ refer to the initial pressure and volume, while P₂ and V₂ refer to the final pressure and volume. To solve for the final volume (V₂), you would rearrange the formula to be V₂ = (P₁V₁) / P₂. So, if you were given the initial pressure and volume, you could substitute those values into the formula, along with the final pressure of 1.6 atm, to find the final volume.

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In this case, the final volume is 31.1 mL.

The final volume of the helium in milliliters can be calculated using Boyle's Law, which states that the product of pressure and volume is constant for a given amount of gas at constant temperature. Initially, the pressure is 447 torr, and the volume is 86.4 mL. When the pressure increases to 1,240 torr, the final volume can be found using the formula:

P1V1 = P2V2

Solving for the final volume, V2, we get:

V2 = (P1V1) / P2

Substitute the given values:

V2 = (447 torr× 86.4 mL) / 1,240 torr

= 31.1 mL

Answer:

a) Temperatura, b) Temperature, c)    Constant , d)  None of these , e) Gibbs enthalpy and free energy (G)

Explanation:

a) the expression for ideal gases is PV = nRT

     Temperature

b) The internal energy is E = K T

      Temperature

c)  S = ΔQ/T

In an isolated system ΔQ is zero, entropy  is constant

       Constant

d) all parameters change when changing status

        None of these

e) Gibbs enthalpy and free energy

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).

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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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:

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:

a) [tex]h=250\ m[/tex]

b) [tex]\Delta h=0.0835\ m[/tex]

Explanation:

Given:

a)

Maximum height reached by the helicopter:

using the equation of motion,

[tex]h=u.t+\frac{1}{2} a.t^2[/tex]

where:

u = initial velocity of the helicopter = 0 (took-off from ground)

t = time of observation

[tex]h=0+0.5\times 5\times 10^2[/tex]

[tex]h=250\ m[/tex]

b)

height fallen freely by Austin:

[tex]h_f=u.t_f+\frac{1}{2} g.t_f^2[/tex]

where:

[tex]u=[/tex] initial velocity of fall at the top = 0 (begins from the max height where the system is momentarily at rest)

[tex]t_f=[/tex] time of free fall

[tex]h_f=0+0.5\times 9.8\times 7^2[/tex]

[tex]h_f=240.1\ m[/tex]

Velocity just before opening the parachute:

[tex]v_f=u+g.t_f[/tex]

[tex]v_f=0+9.8\times 7[/tex]

[tex]v_f=68.6\ m.s^{-1}[/tex]

Time taken by the helicopter to fall:

[tex]h=u.t_h+\frac{1}{2} g.t_h^2[/tex]

where:

[tex]u=[/tex] initial velocity of the helicopter just before it begins falling freely = 0

[tex]t_h=[/tex] time taken by the helicopter to fall on ground

[tex]h=[/tex] height from where it falls = 250 m

now,

[tex]250=0+0.5\times 9.8\times t_h^2[/tex]

[tex]t_h=7.1429\ s[/tex]

From the above time 7 seconds are taken for free fall and the remaining time to fall with parachute.

remaining time,

[tex]t'=t_h-t_f[/tex]

[tex]t'=7.1428-7[/tex]

[tex]t'=0.1428\ s[/tex]

Now the height fallen in the remaining time using parachute:

[tex]h'=v_f.t'+\frac{1}{2} a_p.t'^2[/tex]

[tex]h'=68.6\times 0.1428+0.5\times 2\times 0.1428^2[/tex]

[tex]h'=9.8165\ m[/tex]

Now the height of Austin above the ground when the helicopter crashed on the ground:

[tex]\Delta h=h-(h_f+h')[/tex]

[tex]\Delta h=250-(240.1+9.8165)[/tex]

[tex]\Delta h=0.0835\ m[/tex]

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:

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]

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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]

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]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:

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:

B

Explanation:

The electroscope device is used to detect the presence of charge and it's relative amount. If a charged object is brought near the top of the electroscope(for a example positive charge) the leaves at the bottom spread apart. The leaves diverge further: The positive charge on the leaves has increased further. This happens when positive charge is produced on the leaves by the charged object. This is possible when the object is positively charged.The greater the charge, the farther apart they move.

Answer:

Explanation:

An electroscope is a device used to detect the presence of electric charges. However, to detect charge in an object, it requires hundreds of volts, that's why is only used with high voltage sources.

An important matter is that an electroscope follows the Coulomb electrostatic force, which is a law of physics that quantifies the amount of force between two charged particles, which are stationary.

Having said that, when two charged particles have the same nature, then they will spread apart. Same nature means both negative or both positive.

Therefore, the right answer is B.

To answer this question it is necessary to use the Bohr Model as a theoretical reference. According to this model, the energy levels of an atom are divided by discrete and characteristic energies. When there is the emission or absorption of a photon with a characteristic energy there is a transition of another energy level, which is equal to the level of atomic energy. Since the energy of a photo is directly proportional to its frequency, the emitted or absorbed photons have characteristic frequencies to the difference in energy between atomic energy levels.

Final answer:

The bulk specific gravity (SSD) of the sand is 0.221.

Explanation:

The bulk specific gravity (SSD) of the sand can be calculated using the given information. The combined weight of the sand and water is 1697 g, and the weight of the jar filled with water only is 1390 g. To find the weight of the sand in the SSD condition, we subtract the weight of the jar filled with water from the combined weight: 1697 g - 1390 g = 307 g. Therefore, the bulk specific gravity (SSD) of the sand is the weight of the sand divided by the weight of an equal volume of water, which is 307 g / 1390 g = 0.221.

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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Complete Question

The complete question is shown on the first uploaded image

Answer:

  The dopant density is ND   ≈  8.135*10¹²  cm⁻³

Explanation:

The explanation is shown on the second , third ,fourth and fifth uploaded image