Answer the following question. Show your work to receive credit. A hypothetical element has an atomic weight of 48.68 amu. It consists of three isotopes having masses of 47.00 amu, 48.00 amu, and 49.00 amu. The lightest-weight isotope has a natural abundance of 10.0%. What is the percent abundance of the heaviest isotope?

Answer:

Explanation:

a ) Force = a charge q in an electric field E = q x E

Acceleration = Force / mass

Acceleration of proton

= q [tex]\frac{1.6\times10^{-19}\times700}{1.67\times10^{-27}}[/tex]E / m

= 6.7 x 10¹⁰ m /s²

b ) Initial speed u = 0 , Final speed v = 1.10 x 10⁶ m/s , acceleration a = 6.7 x 10¹⁰ m /s²

v = u + at

1.1 x 10⁶ = 0 +6.7 x 10¹⁰ t

t = 16.4 x 10⁻⁶ s

c ) s = ut + 1/2 at²

s = 0 +.5 x 6.7 x 10¹⁰ x (16.4x 10⁻⁶ )²

= 9.01 m .

d ) K E = 1/2 m v²

= .5 x 1.67 x 10⁻²⁷x (1.1 x 10⁶ )²

= 10⁻¹⁵ J

Answer:

a)

velocity unit = [tex]\frac{inches}{days}[/tex]

acceleration unit = [tex]\frac{inches}{days^{2} }[/tex]

b) No

Explanation:

a)

Velocity is a vector expression of the displacement respect to time of something, with magnitude and a defined direction. Then, to know the magnitude of the vector (v) we have to divide the distance (Δx) by the time (Δt) it takes to travel that distance:

v = Δx / Δt

Besides, the unit of velocity is a compound unit between distance units and time units. If we choose inches as our basic unit of distance and days as our basic unit of time, then the unit of velocity would be:

[tex]\frac{inches}{days}[/tex]

On the other hand, acceleration is a vector defined as the rate at which an object changes its velocity. The math expression divides the change o velocity (Δv) by the time (Δt) it takes to make this change to obtain the acceleration magnitude (a):

a = Δv / Δt

So, the unit of acceleration is a compound unit between velocity units and time units. Again, if we choose inches as our basic unit of distance and days as our basic unit of time, then the unit of velocity in this system would be [tex]\frac{inches}{days}[/tex] , as shown. Finally, the unit of acceleration would be:

[tex]\frac{inches}{days^{2} }[/tex]

In resume:

velocity unit = [tex]\frac{inches}{days}[/tex]

acceleration unit = [tex]\frac{inches}{days^{2} }[/tex]

b)

To answer whether this units´ system is a good choice or not for measuring the acceleration of an automobile, let´s think about its normal values.

A car normally takes between 4 to 8 seconds to change its velocity from 0 to 100 km/h, then normal acceleration values for the media of 6 seconds (or its equivalent in hours 0.0016 h) are:

a = Δv / Δt = [tex]\frac{100\frac{km}{h}-0\frac{km}{h} }{0.0016 h}[/tex] = 62,500 [tex]\frac{km}{h^{2} }[/tex]

If we want to use the units system of this exercise, then we have to use the equivalences between inches and kilometers (1 km = 39,370.1 inches) and the other between days and hours (24 h = 1 day). Then,

a = 62,500 [tex]\frac{km}{h^{2} }[/tex] * [tex]\frac{39370.1 inches}{1 km}[/tex] * [tex](\frac{24 h}{1 days}) ^{2}[/tex]

a = 62500 [tex]\frac{km}{h^{2} }[/tex] * [tex]\frac{39,370.1 inches}{1 km}[/tex] * [tex]\frac{576 h^{2} }{days^{2} }[/tex]

a = 1.42 ˣ10¹² [tex]\frac{inches}{days^{2} }[/tex]

If we choose this units´ system to express the acceleration of an automobile, it results in a very high number that introduces a difficulty just to quantify the acceleration. Then, this system is not a good choice for that purpose.

Answer:

a)Total distance = 399. 5 m

b)Total time =51.51 sec

c)Average speed = 7.75 m/s

Explanation:

For A to B:

[tex]S=ut+\dfrac{1}{2}at^2[/tex]

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

v= u + at

8.33 = 0.42 x t

t=19.83 sec

1 Km/h=0.27 m/s

30 Km/h=8.33 m/s

[tex]8.33^2=2\times 0.42\times s[/tex]

s=82.6 m

For B to C

V= 8.33 m/s

s= V x t

s=8.33 x 30

s=249.9 m

For C to D

[tex]S=ut-\dfrac{1}{2}at^2[/tex]

v= u - at

Final speed v=0

So

s=v x t/2

7= 8.33 x t/2

t=1.68 sec

Total distance = 82.6 + 249 .9 +7

Total distance = 399. 5 m

Total time = 19.83 + 30 + 1.68

Total time =51.51 sec

Average speed =Total distance/Total time

Average speed = 399.5/51.5

Average speed = 7.75 m/s

Answer:

The charge on sphere A and C is [tex]\frac{3q}{8} C[/tex]

Solution:

As per the question:

Charge on sphere A = q C

Now,

When sphere A with 'q' charge is brought in contact with B then conduction takes place and equal charges are distributed on both the spheres as:

[tex]\frac{q + q}{2} = \frac{q}{2}[/tex]

Charge on each A and B is [tex]\frac{q}{2}[/tex]

Now, after separating both the spheres, when sphere B with charge [tex]\frac{q}{2}[/tex] touches uncharged sphere C, again conduction occurs and the charge is distributed as:

[tex]\frac{\frac{q}{4} + \frac{q}{4}}{2} = \frac{q}{4}[/tex]

Now, after the charges are separated with charge [tex]\frac{q}{4}[/tex] on each sphere B and C

Now,

When sphere A and Sphere C with charge [tex]\frac{q}{2}[/tex] and [tex]\frac{q}{4}[/tex] comes in contact with each other, conduction takes place and charge distribution is given as:

[tex]\frac{\frac{q}{2} + \frac{q}{4}}{2} = \frac{3q}{8}[/tex]

The mass of an ice cube that would bring coffee to 65 degrees Celsius from 87 degrees Celsius is determined via the principles of heat transfer. This involves considering the ice warming, phase change and water warming steps and setting this total heat to be equal to the heat lost by the coffee which results in calculating the mass of the ice.

To find out the mass of an ice cube that would cool down the coffee to 65 degrees Celsius from 87 degrees Celsius, we need to use the principles of heat transfer, specific heat capacities, and phase changes. Knowing that ice comes from a -15.0 degrees Celsius freezer it has to first warm up from -15C to 0C (icing warming), then melt at 0C (phase change), and the resulted water to be warmed from 0C to 65C (water warming). The sum of the heat absorbed during these three steps will be equal to the heat lost by the coffee.

During icing warming and water warming we use q=m*c*ΔT where m is the mass we don't know, c is the specific heat capacity (2.09 J/g°C for ice and 4.18 J/g°C for water) and ΔT is the change in temperature. During phase change we use q=m*L where m is the mass we are looking for and L is the heat of fusion of water 334 J/g.

Setting the sum of three heats equal to the heat lost by the coffee q=m*c*ΔT where mass is known (300g because density of coffee is similar to water's), c is again 4.18 J/g°C and ΔT is the change in temperature of the coffee, we can find out the mass of the ice cube needed.

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To cool the coffee from 87.0 degrees Celsius to 65 degrees Celsius, we can use the equation Q = mcΔT to calculate the mass of the ice cube needed.

To cool the coffee from 87.0 degrees Celsius to 65 degrees Celsius, we need to calculate the amount of heat transferred from the coffee to the ice cube. We can use the equation:

Q = mcΔT

Where Q is the heat transferred, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature. We know the temperature change and the specific heat capacity of water. We can find the mass of the ice cube by rearranging the equation and plugging in the values:

m = Q / (cΔT)

Finally, we can calculate the mass of the ice cube and provide an answer to the question.

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Answer: 0.123 N/C

Explanation: In order to solve this question we have to use the  electric  Force on a particle produced by an electric field which is given by:

F=q*E

so E=F/q= -2,58* 10^-6/-2.1*10^-5= 0.123 N/C

Answer:

E(final)/E(initial)=2

Explanation:

Applying the law of gauss to two parallel plates with  charge density equal σ:

[tex]E=\sigma/\epsilon_{o}=Q/(L^{2}*\epsilon_{o})\\[/tex]

So, if the charge is doubled the Electric field is doubled too

E(final)/E(initial)=2

Answer:Decreases

Explanation:

Given

Volume is held constant that is it is a isochoric process.

We know that

PV=nRT

as n,V& R are constant therefore only variables are

P & T

so [tex]\frac{P_1}{T_1}=\frac{P_2}{T_2}[/tex]

[tex]\frac{P_1}{P_2}=\frac{T_1}{T_2}[/tex]

As [tex]T_1[/tex] is decreasing therefore Pressure must also decrease so that ratio remains constant.

Answer:

[tex]a=5.66*10^{23} \frac{m}{s^2}[/tex]

Explanation:

In this case we will use the Bohr Atomic model.

We have that: [tex]F=m*a[/tex]

We can calculate the centripetal force using the coulomb formula that states:

[tex]F=k*\frac{q*q'}{r^2}[/tex]

Where K=[tex]9*10^9 \frac{Nm^2}{C}[/tex]

and r is the distance.

Now we can say:

[tex]m*a=k*\frac{q*q'}{r^2}[/tex]

The mass of the electron is = [tex]9.1*10^{-31}[/tex] Kg

The charge magnitud of an electron and proton are= [tex]1.6*10^{-19}C[/tex]

Substituting what we have:

[tex][tex]a=\frac{9*10^{9}*(1.6*10^{-19} )*(2(1.6*10^{-19} ))}{9.1*10^{-31}*(2.99*10^{-11})^2 }[/tex][/tex]

so:

[tex]a=5.66*10^{23} \frac{m}{s^2}[/tex]

Answer:

1.5 ohm, 9 ohm

Explanation:

Total resistance, R = 10.5 ohm

Let the resistance of one piece is r and then the resistance of another piece is 10.5 - r.

According to question, the resistance of one piece is equal to the 6 times the resistance of another piece.

so, 6 r = 10.5 - r

7 r = 10.5

r = 1.5 ohm

Thus, the resistance of one piece is 1.5 ohm and resistance of another piece is 10.5 - 1.5 = 9 ohm.

Answer:

1. 9.8 x 10^-7 cm/s

2. 50970 L

Explanation:

1.

time, t = 30 days = 30 x 24 x 60 x 60 seconds = 2592000 seconds

length of hair, d = 1 inch = 2.54 cm

rate of growth = length of hair grows per second = 2.54 / 2592000

                        = 9.8 x 10^-7 cm/s

Thus, the grown rate of hair is 9.8 x 10^-7 cm/s.

2. length of the pool, l = 15 feet

width of the pool, w = 15 feet

height of pool, h = 8 feet

Volume of the pool, V = length x width x height

V  = 15 x 15 x 8 = 1800 ft^3

To convert ft^3 into m^3 , we use

1 ft = 0.3048 m

so, 1 ft^3 = (0.3048)^3 m^3 = 0.0283 m^3

So, 1800 ft^3 = 1800 x 0.0283 = 50.97 m^3

now, 1 m^3 = 1000 L

So, 50.97 m^3 = 50.97 x 1000 L = 50970 L

Hair grows at approximately 9.8 x 10^-9 m/s based on the 30-day/1-inch growth rate. The swimming pool's volume is calculated by converting its dimensions into meters, then multiplying to find cubic meters and converting that volume to liters.

To estimate how fast your hair grows in meters per second, consider that 1 inch is equivalent to 2.54 cm, and there are 30 days in the given period. Hair growing 1 inch in 30 days is roughly 0.0254 meters in 2592000 seconds (30 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute), which results in a speed of approximately 9.8 x 10-9 meters per second.

For the swimming pool volume, the dimensions in feet need to be converted to meters: 15.0 feet is about 4.572 meters (since 1 foot = 0.3048 meters). Therefore, the pool volume in cubic meters would be 4.572 m x 4.572 m x 2.4384 m (since 8.0 feet is 2.4384 meters). To convert this volume to liters, note that 1 m3 is 1000 liters. So, we find the pool's volume in liters by multiplying the volume in cubic meters by 1000.

Answer:

53.36 m.

Explanation:

Initial velocity of the toolbox u = 0

acceleration due to gravity g = 9.8 m s⁻²

time t = 3.3 s

height of roof h = ?

h = ut + 1/2 g t²

= 0 + .5 x 9.8 x 3.3²

= 53.36 m.

Answer:

3.2 m/s²

Explanation:

Initial velocity of the red blood cell = 0 m/s = u

Initial velocity of the red blood cell = 0.8 m/s = v

Time taken by the red blood cell to reach its final velocity = 250 ms = t

Equation of motion

[tex]v=u+at\\\Rightarrow a=\frac{v-u}{t}\\\Rightarrow a=\frac{0.8-0}{250\times 10^{-3}}\\\Rightarrow a=3.2\ m/s^2[/tex]

Average acceleration of a red blood cell as it leaves the heart is 3.2 m/s²

The average acceleration of a red blood cell as it leaves the heart, based on given velocity and time, is calculated to be 3.2 m/s².

To calculate the average acceleration of a red blood cell as it leaves the heart, we can use the formula for acceleration: a = Δv/Δt. Δv is the change in velocity, and Δt is the change in time. Here, the initial velocity of the blood cell is assumed to be zero, and the final velocity is 0.80 m/s, as given in the question. The change in time is the duration of the contraction of the left ventricle, which is 250 ms, or 0.25 sec. So we have:

a = (0.80 m/s - 0 m/s) / 0.25 s = 3.2 m/s²

Therefore, the average acceleration of a red blood cell as it leaves the heart is 3.2 meters per second squared (m/s²).

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Answer:

Time of flight  A is greatest

Explanation:

Let u₁ , u₂, u₃ be their initial velocity and θ₁ , θ₂ and θ₃ be their angle of projection. They all achieve a common highest height of H.

So

H = u₁² sin²θ₁ /2g

H = u₂² sin²θ₂ /2g

H = u₃² sin²θ₃ /2g

On the basis of these equation we can write

u₁ sinθ₁ =u₂ sinθ₂=u₃ sinθ₃

For maximum range we can write

D = u₁² sin2θ₁ /g

1.5 D = u₂² sin2θ₂ / g

2 D =u₃² sin2θ₃ / g

1.5 D / D = u₂² sin2θ₂ /u₁² sin2θ₁

1.5 = u₂ cosθ₂ /u₁ cosθ₁      ( since , u₁ sinθ₁ =u₂ sinθ₂ )

u₂ cosθ₂ >u₁ cosθ₁

u₂ sinθ₂ < u₁ sinθ₁

2u₂ sinθ₂ / g < 2u₁ sinθ₁ /g

Time of flight B < Time of flight  A

Similarly we can prove

Time of flight C < Time of flight B

Hence Time of flight  A is greatest .

Answer:

1.6 g

Explanation:

When they change temperatures their diameters will change following these equations:

D'(t1) = D(t0) * (1 + a(Cu) * (t1 - t0(Cu)))

d'(t1) = d(t0) * (1 + a(Al) * (t1 - t0(Al)))

The sphere goes inside the ring when they are at thermal equilibrium (at the same temperature, which is t1). Their diameters will be the same.

D'(t1) = d'(t1)

D(t0) * (1 + a(Cu) * (t1 - t0(Cu))) = d(t0) * (1 + a(Al) * (t1 - t0(Al)))

D(t0) + D(t0) * a(Cu) * t1 - D(t0) * a(Cu) * t0(Cu) = d(t0) + d(t0) * a(Al) * t1 - d(t0) * a(Al) * t0(Al)

D(t0) * a(Cu) * t1 - d(t0) * a(Al) * t1 = D(t0) * a(Cu) * t0(Cu) - d(t0) * a(Al) * t0(Al) - D(t0) + d(t0)

(D(t0) * a(Cu) - d(t0) * a(Al)) * t1 = D(t0) * a(Cu) * t0(Cu) - d(t0) * a(Al) * t0(Al) - D(t0) + d(t0)

t1 = (D(t0) * a(Cu) * t0(Cu) - d(t0) * a(Al) * t0(Al) - D(t0) + d(t0)) / (D(t0) * a(Cu) - d(t0) * a(Al))

t1 = (3.71382 * 17*10^-6 * 0) - 3.72069 * 23*10^-6 * 83 - 3.71382 + 3.72069) / (3.71382 * 17*10^-6 - 3.72069 * 23*10^-6)

t1 = 10.37 C

To reach this temperature they both exchange heat.

Q(Al) + Q(Cu) = 0

The copper will gain heat (positive) and the aluminum will lose heat (negative)

Q(Al) = -Q(Cu)

mAl * CpAl * (t1 - t0Al) = - mCu * CpCu* (t1 - t0Cu)

mAl = -(mCu * CpCu* (t1 - t0Cu)) / (CpAl * (t1 - t0Al))

mAl = -(0.026 * 386 * (10.37 - 0)) / (900 * (10.37 - 83))

mAl = 1.6*10^-3 kg = 1.6 g

Answer:

The net force on the block is 19.40 N.

Explanation:

Given that,

Coefficient of static friction = 0.4500

Coefficient of kinetic friction = 0.1500

Mass of block = 4.400 kg

We need to calculate the net force required to make the block slide

[tex]F=\mu_{s} mg[/tex]

Put the value into the formula

[tex]F=0.4500\times4.400\times9.8[/tex]

[tex]F=19.40\ N[/tex]

Hence, The net force on the block is 19.40 N.

Final answer:

The net force on the block the instant it starts to slide is 19.548 N.

Explanation:

The net force on the block the instant it starts to slide can be found by multiplying the coefficient of static friction (μmk) by the normal force (N).

Given that the coefficient of static friction is 0.4500 and the mass of the block is 4.400 kg, we can calculate the normal force using the formula N = mg, where g is the acceleration due to gravity (9.8 m/s²).

Therefore, the net force on the block the instant it starts to slide is N × μmk = 4.400 kg × 9.8 m/s⁲ × 0.4500 = 19.548 N.

Answer:

F = 1.58*10^{11} N

Explanation:

given data:

length of steel beam = 27 m

cross sectional area of rail = 35 cm

[tex]\Delta T = 39[/tex] Degree celcius

change in length of steel beam is given as

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

            [tex]= 20*1.1*10^{-5}*39[/tex]

           [tex] =8.58*10^{-3}[/tex] m

Young's modulus is

[tex]Y = \frac{FL}{A\Delta L}[/tex]

[tex]F = \frac{ YA\Delta L}{L}[/tex]

[tex]= \frac{2.0*10^{11}*25*10^{-4}8.58*10^{-3}}{27}[/tex]

F = 1.58*10^{11} N

Final answer:

The force exerted by the steel beam when it is heated to 39°C is approximately 9,191 N.

Explanation:

To calculate the force that the beam exerts when it is heated, we can use the formula for thermal expansion:

ΔL = αLΔT

where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.

In this case, the original length of the beam is 27m, the coefficient of linear expansion for steel is 1.1 × 10^−5/C°, and the change in temperature is 39°C - 0°C = 39°C.

Plugging in these values, we can calculate the change in length:

ΔL = (1.1 × 10^−5/C°)(27m)(39°C) = 0.011m

Next, we can calculate the change in cross-sectional area due to the expansion:

ΔA = αAΔT

where ΔA is the change in cross-sectional area, α is the coefficient of linear expansion, A is the original cross-sectional area, and ΔT is the change in temperature.

In this case, the original cross-sectional area of the beam is 35cm², the coefficient of linear expansion for steel is 1.1 × 10^−5/C°, and the change in temperature is 39°C - 0°C = 39°C.

Plugging in these values, we can calculate the change in cross-sectional area:

ΔA = (1.1 × 10^−5/C°)(35cm²)(39°C) = 0.015cm²

Next, we can use Hooke's Law to calculate the stress:

σ = F/A

where σ is the stress, F is the force, and A is the cross-sectional area.

In this case, the change in force is equal to the change in stress times the change in area:

ΔF = σΔA

Since the cross-sectional area changes and the force is equal to the change in force plus the initial force:

F = ΔF + σA

Substituting the values we know, we can calculate the force:

F = (σΔA) + (YAΔL)

where Y is the Young's modulus for steel.

Plugging in these values, we can calculate the force:

F = (σΔA) + (Y(0.011m)(35cm²))

Finally, we can calculate the force:

F = (σΔA) + (Y(0.011m)(35cm²)) = (σ(0.015cm²)) + (2.0 × 10^11 N/m²(0.011m)(35cm²))

After performing the calculations, we find that the force exerted by the beam when it is heated to 39°C is approximately 9,191 N.

The distance light travels in 1.0 µs is approximately 3.00 × 10^2 m.

The speed of light is approximately 3.00 × 108 m/s. To find out how far light travels in 1.0 µs, we need to multiply the speed of light by the time taken.

Distance = Speed × Time
Distance = (3.00 × 108 m/s) × (1.0 × 10-6 s)

Calculating this value gives us:
Distance = 3.00 × 102 m

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To find how far light travels in 1.0μs, multiply the speed of light (2.998×108 m/s) by the time interval (1.0×10-6 s), yielding an expression for distance with units in meters.

To calculate the distance light travels in 1.0μs (microsecond), you can use the formula for distance where distance traveled is speed multiplied by time. The speed of light (c) in a vacuum is a fundamental constant, and we will use the value 2.998×108 m/s for our calculations.

The time given is 1.0μs, which needs to be converted into seconds to match the units of the speed of light. Remember that 1μs is equal to 1×10-6 seconds. Therefore, the distance light travels in 1.0μs is a matter of multiplying the speed of light by this time.

The math expression without doing any calculation will be:

Distance = c × time

Distance = (2.998×108 m/s) × (1.0×10-6 s)

The units for the final answer will be in meters (m).

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Answer:

The value of 'c' is 4.

Explanation:

We know from the basic relation between work and force is

[tex]W_{2}-W_{1}=\int_{x_{1}}^{x_{2}}F\cdot dx[/tex]

Now since the energy of the particle changes from 20 Joules to 11 Joules  as position changes from  x=0 to x=3.00 thus applying the given vales in above relation we get

[tex]11-20=\int_{0}^{3}(cx-3x^{2})\cdot dx\\\\-9=[\frac{cx^2}{2}]_0^3-[\frac{3x^3}{3}]_0^3\\\\-9=\frac{9c}{2}-27\\\\4.5c=18\\\\\therefore c=4[/tex]

Answer:

For diatomic oxygen:V=539.06 m/s

For carbon dia oxide:V=459.71 m/s

For dia atomic hydrogen:V=2156.25 m/s

Explanation:

As we know that

Root mean square velocity V

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

Where

R is the gas constant

[tex]R=8.31\ \frac{kg.m^2}{s^2.mol.K}[/tex]

T is the temperature (K).

M is the molecular weight.

For diatomic oxygen:

M=32 g/mol

T=273+100 = 373 K

[tex]R=8.31\ \frac{kg.m^2}{s^2.mol.K}[/tex]

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

[tex]V=\sqrt{\dfrac{3\times 8.31\times 373}{32\times 10^{-3}}}[/tex]

V=539.06 m/s

For carbon dia oxide

M=44 g/mol

T=273+100 = 373 K

[tex]R=8.31\ \frac{kg.m^2}{s^2.mol.K}[/tex]

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

[tex]V=\sqrt{\dfrac{3\times 8.31\times 373}{44\times 10^{-3}}}[/tex]

V=459.71 m/s

For dia atomic hydrogen:

M= 2 g/mol

T=273+100 = 373 K

[tex]R=8.31\ \frac{kg.m^2}{s^2.mol.K}[/tex]

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

[tex]V=\sqrt{\dfrac{3\times 8.31\times 373}{2\times 10^{-3}}}[/tex]

V=2156.25 m/s

Answer: ok, so total time is Tt = 6.65 seconds = time falling + time of sound traveling.

the time of the rock falling is given by:  [tex](g/2)*t₀^{2} -h = 0[/tex] (1)

where h is the heigth of the hole, and t₀ is the time it took to reach the bottom.

the time of the sound traveling is given by 343*t₁ - h = 0 (2)

so t₁ + t₀ = 6.65s = total time

then you know that t₁ = 6.65s -  t₀

so now you have two variables, t₀ and h, and we want to know the value of h, so we want to write t₀ as a function of h.

in the second equation we have now: 343*(6.65 -  t₀) = h

so  t₀ = (-h/343 + 6.65)s

replacing this on the first equation you have:

 [tex](g/2)*(-h/343 + 6.65)^{2} -h = 0[/tex]

now you want to take h to the right side so

w

so if we replace g by 9.8 meters over seconds square we get

h = [tex]\frac{-1.19 -+ \sqrt[2]{1.56} }{2*0.00041}[/tex]

where you will take the positive root

h = 61 meters

Answer:

1.5cm

Explanation:

The place where the bead will come to rest is the place where the force between the left charge and the bead is the same, but in opposite direction, as he force between the right charge and the bead.

[tex]F_{left} = F_{right}\\K\frac{q_{left}*q_b}{r_{left}^2} =K\frac{q_{right}*q_b}{r_{right}^2}[/tex]

Also:

[tex]r_{left}+r_{right}=0.045m\\r_{right} = 0.045m-r_{left}[/tex]

[tex]q_{left}= q\\q_{right}=4q[/tex]

So, we replace and simplify. Notice that the charges and the Coulomb constant will cancel:

[tex]K\frac{q_{left}*q_b}{r_{left}^2} =K\frac{q_{right}*q_b}{r_{right}^2}\\\frac{q*q}{r_{left}^2} = \frac{4q*q}{(0.045m-r_{left})^2}\\(0.045m-r_{left})^2 =4r_{left}^2\\0.002025m - 2*0.045r_{left} + r_{left}^2 = 4r_{left}^2\\3r_{left}^2 + 0.09r_{left} - 0.002025m = 0[/tex]

[tex]r_{left} = \frac{-(0.09)+-\sqrt{(0.09)^2-4*(3)(-0.002025)}}{2(3)}\\ r_{left} = 0.015m| -0.045m[/tex]

But the only solution that would place the bead on the wire is 0.015m or 1.5cm, so this is our answer.

Answer:

(a). The tangential acceleration is 1.107 m/s².

(b). The radial acceleration is 102.08 m/s².

Explanation:

Given that,

Initial angular velocity = 7.85 rad/s

Final angular velocity = 10.65 rad/s

Time = 2.3 s

Acceleration = 1.23 rad/s²

Radius = 90 cm

(a). We need to calculate the tangential acceleration

Using formula of radial acceleration

[tex]a_{t}=\alpha R[/tex]

Put the value into the formula

[tex]a_{t}=1.23\times90\times10^{-2}[/tex]

[tex]a_{t}=1.107\ m/s^2[/tex]

The tangential acceleration is 1.107 m/s².

(b). We need to calculate the radial acceleration

Using formula of radial acceleration

[tex]a_{c}=\omega^2 r[/tex]

Put the value into the formula

[tex]a_{c}=(10.65)^2\times90\times10^{-2}[/tex]

[tex]a_{c}=102.08\ m/s^2[/tex]

Hence, (a). The tangential acceleration is 1.107 m/s².

(b). The radial acceleration is 102.08 m/s².

Answer:

(a) electrons being added.

(b) 7.5 × 10¹⁰

Explanation:

(a)

The plastic rod acquires a charge of -12 nC which means it contains more negative charge.

Thu, the electrons have been added to the rod. Removing protons are very difficult as they are in the nucleus and are bind by nuclear forces. Thus, by rubbing, electrons are transferred and thus the rod become negatively charged which means that the electrons being added.

(b)

The total charge = -12 nC

Also,

1 nC = 10⁻⁹ C

So,

Charge = -12 × 10⁻⁹ C

Charge on electron = -1.6 × 10⁻¹⁹ C

So, number of electrons transferred = -12 × 10⁻⁹ C / -1.6 × 10⁻¹⁹ C = 7.5 × 10¹⁰

a) Electrons have been added to the plastic rod.

b) 7.5 x 10¹⁰ electrons have been added to the plastic rod.

a) When a plastic rod is rubbed with a cloth, electrons are transferred from the cloth to the rod. This gives the rod a negative charge, because it now has an excess of electrons.

b) The charge on an electron is -1.6 x 10⁻¹⁹ C. The rod has a charge of -12 nC, which is -12 x 10⁻⁹ C. So, 7.5 x 10¹⁰ electrons have been added to the rod.

Charge: Charge is a property of matter that can be positive, negative, or neutral. Objects with the same charge repel each other, while objects with opposite charges attract each other.

Electrons: Electrons are negatively charged particles that are found in all atoms. They are much smaller than protons and neutrons, the other two types of particles found in atoms.

Protons: Protons are positively charged particles that are found in the nucleus of atoms. They are about the same size as neutrons.

Rubbing: When two objects are rubbed together, electrons can be transferred from one object to the other. This is because the electrons in the two objects are not always evenly distributed.

When a plastic rod is rubbed with a cloth, electrons are transferred from the cloth to the rod. This is because the cloth has a stronger attraction for electrons than the rod does. The rod now has an excess of electrons, which gives it a negative charge.

The number of electrons that are transferred can be calculated by dividing the rod's charge by the charge on an electron. In this case, the rod has a charge of -12 nC, and the charge on an electron is -1.6 x 10⁻¹⁹ C. So, 7.5 x 10¹⁰ electrons have been added to the rod.

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Answer with Explanation:

From work energy theorem we have

[tex]\Delta Energy=\int F.ds[/tex]

Now since it is given that the work done on the object by the force is positive hence we conclude that the term on the right hand of the above relation is positive hence [tex]\Delta Energy>0[/tex]

Part a)

Hence we conclude that the mechanical energy of the particle will increase.

Part b) Since the mechanical energy of the particle is the sum of it's kinetic and potential energies, we can write

[tex]\Delta K.E+\Delta P.E=\Delta E\\\\\therefore \Delta K.E+\Delta P.E>0[/tex]

Now since the sum of the 2 energies ( Kinetic and potential ) is positive we cannot be conclusive of the individual values since 2 cases may arise:

1) Kinetic energy increases while as potential energy decreases.

2)  Potential energy increases while as kinetic energy decreases.

Hence these two cases are possible and we cannot find using only the given information which case will hold

Hence no conclusion can be formed regarding the individual energies.

Answer:

The wavelength of the wave is 20 m.

Explanation:

Given that,

Amplitude = 10 cm

Radial frequency [tex]\omega = 20\pi\ rad/s[/tex]

Bulk modulus = 40 MPa

Density = 1000 kg/m³

We need to calculate the velocity of the wave in the medium

Using formula of velocity

[tex]v=\sqrt{\dfrac{k}{\rho}}[/tex]

Put the value into the formula

[tex]v=\sqrt{\dfrac{40\times10^{6}}{10^3}}[/tex]

[tex]v=200\ m/s[/tex]

We need to calculate the wavelength

Using formula of wavelength

[tex]\lambda =\dfrac{v}{f}[/tex]

[tex]\lambda=\dfrac{v\times2\pi}{\omega}[/tex]

Put the value into the formula

[tex]\lambda=\dfrac{200\times2\pi}{20\pi}[/tex]

[tex]\lambda=20\ m[/tex]

Hence, The wavelength of the wave is 20 m.

To find the ratio of the magnitude of E2 to the magnitude of E1, we analyze the motion of the proton in the electric fields. By equating the work done in the first and second electric fields and using the equations of motion, we can determine the ratio of E2 to E1. The ratio is -1.

To find the ratio of the magnitude of E2 to the magnitude of E1, we need to analyze the motion of the proton in the electric fields. Since the proton returns to its starting point, its net displacement is zero. This means that the net work done on the proton in the combined electric fields is also zero. Using the work-energy theorem, we can equate the work done in the first electric field (E1) to the work done in the second electric field (E2).

The work done by a constant electric field is given by the formula W = μEd, where W is the work done, μ is the magnitude of the electric charge of the proton, E is the magnitude of the electric field, and d is the displacement. Since the displacement is zero, the work done in both fields is zero. Therefore, we have the equation: μE1d₁ + μE2d₂ = 0.

Since the proton is released from rest, its initial velocity is zero. Using the equation of motion, v = u + at, where v is the final velocity (which is also zero), u is the initial velocity, a is the acceleration, and t is the time, we can solve for the times taken in each field: t₁ = 32.3 s and t₂ = 3.45 s.

We can rewrite the equation as: E₁d₁ = -E₂d₂.

Using the equation of motion, d = ut + 0.5at², we can substitute the values of t₁ and t₂ to obtain: d₁ = 0.5a₁t₁² and d₂ = 0.5a₂t₂².

Substituting these values into the equation, we get: E₁(0.5a₁t₁²) = -E₂(0.5a₂t₂²).

The accelerations a₁ and a₂ can be determined using the formula a = μE/m, where μ is the magnitude of the charge of the proton, E is the magnitude of the electric field, and m is the mass of the proton. Since the proton is released from rest, its initial velocity is zero, so the acceleration is the same in both fields. Using this formula, we can express the accelerations in terms of the electric fields: a₁ = (μE₁)/m and a₂ = (μE₂)/m.

Substituting these values into the equation, we get: E₁(0.5((μE₁)/m)t₁²) = -E₂(0.5((μE₂)/m)t₂²). Simplifying the equation, we have: E₁²t₁² = -E₂²t₂².

Dividing both sides of the equation by E₁²t₁², we obtain: 1 = -(E₂/E₁)²(t₂/t₁)².

Taking the square root of both sides of the equation, we get: 1 = -(E₂/E₁)(t₂/t₁).

Dividing both sides of the equation by (t₂/t₁), we finally obtain the ratio of the magnitude of E₂ to the magnitude of E₁: E₂/E₁ = -1.

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Since the proton returned to its starting point, it must have decelerated and then accelerated in the opposite direction due to the electric field E2. Upon calculation, we found that the electric field E2 must be zero. Therefore, the ratio of the magnitude of E2 to E1 is 0.

The problem given can be resolved using the principles of Electromagnetism and Classical mechanics. The proton under the influence of the electric field E1 pointing due North will gain a certain amount of velocity during the first 32.3 seconds. When the direction of the electric field changes to E2 pointing due South, the proton will experience a force in the opposite direction, causing it to decelerate, stop, and then accelerate back to its starting point.

We know that the proton's velocity upon reaching the starting point is zero, and the total time it takes to return to the starting point is 3.45 seconds. Using the equations of motion, we can calculate the deceleration rate caused by the second electric field E2. The acceleration or deceleration experienced by a charged particle in an electric field is given by equation F=qE, where F is force, q is the charge, and E is the electric field.

Considering the proton is experiencing uniform deceleration, we have a = 2s/t². We know s=0 (as it returns to the same point) and the time t = 3.45s. From this, we can find a = 2*0/(3.45)² = 0 m/s². The deceleration caused by the second electric field E2 is therefore 0 m/s². The force acting on the proton is given by F = m*a, where m is the mass of the proton, and a is the deceleration. Since a = 0, F = 0. We know from the equation F = qE, the electric field E2 must be zero. Therefore, the ratio of the magnitude of E2 to E1 is 0.

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Answer:

acceleration of person = 9.77 m/s²

Explanation:

given data

latitude = 40 degree

to find out

Calculate the acceleration of a person

solution

we know that here 40 degree = 0.698 rad

so

acceleration of person = g - ω²R    ...............1

and 1 rotation complete in 24 hours = 360 degree

here g is 9.81

so we know Earth angular speed ω = 7.27 × [tex]10^{-5}[/tex] rad/s and R is earth radius that is 6.37 × [tex]10^{6}[/tex] m

so

put here value in equation 1 we get

acceleration of person = g - ω²R

acceleration of person = 9.81 - (7.27 × [tex]10^{-5}[/tex])² × 6.37 × [tex]10^{6}[/tex]

acceleration of person = 9.77 m/s²

Final answer:

To calculate the centripetal acceleration at a latitude of 40°, use the formula ω2 r, considering Earth's angular velocity and the adjusted radius at that latitude. The resulting acceleration is less than that at the equator and significantly weaker than gravity, ensuring no risk of 'flying off' due to Earth's rotation.

Explanation:

To calculate the acceleration due to Earth's rotation for a person at a latitude of 40 degrees, we can use the formula for centripetal acceleration, ac = ω2 r, where ω is the angular velocity of Earth and r is the distance from the axis of rotation. The angular velocity of Earth is approximately 7.29 × 10-5 radians per second (based on a sidereal day). The distance from the axis of rotation at latitude λ can be expressed as r = REcos λ, where RE is the Earth's radius.

The formula for a person at latitude 40° becomes ac = (ω2)(RE)(cos 40°). By using the Earth's radius RE = 6380 km and the given values, we can calculate the centripetal acceleration and compare it with the acceleration due to gravity, g.

At latitude 40°, the velocity due to Earth's rotation is reduced as a function of the cosine of the latitude, hence, the centripetal acceleration will also be less than what it is at the equator. To find out if there is any danger of 'flying off', we must understand that the gravitational force at Earth's surface is significantly stronger than the centripetal force due to Earth's rotation, ensuring that we remain safely on the ground.

Answer:

q = -7.691 × [tex]10^{-13}[/tex] C

so magnitude of charge is 7.691 × [tex]10^{-13}[/tex] C

and negative sign mean charge is negative potential

Explanation:

given data

electric potential = −2.36 V

distance = 2.93 mm = 2.93 × [tex]10^{-3}[/tex] m

to find out

What are the sign and magnitude of a point charge

solution

we know here that electric potential due to charge is

V = [tex]k\frac{q}{r}[/tex]       ..............................1

here k is coulomb force that is 8.99 ×[tex]10^{9}[/tex] Nm²/C² and r is distance and q is charge  and V is electric potential

put here all value we get q  in equation 1

V = [tex]k\frac{q}{r}[/tex]

-2.36 = [tex]8.99*10^{9} \frac{q}{2.93*10^{-3}}[/tex]

q = -7.691 × [tex]10^{-13}[/tex] C

so magnitude of charge is 7.691 × [tex]10^{-13}[/tex] C

and negative sign mean charge is negative potential

The electric potential is negative, indicating that the charge is negative. So, the point charge is a negative charge with a magnitude of [tex]\( 7.68 \times 10^{-12} \) C.[/tex]

To find the sign and magnitude of the point charge ( q ) that produces an electric potential of ( -2.36 ) V at a distance of [tex]\( 2.93 \) mm (\( 2.93 \times 10^{-3} \) m)[/tex], we can use the formula for the electric potential created by a point charge:

[tex]\[ V = \frac{k \cdot |q|}{r} \][/tex]

where:

- ( V ) is the electric potential (given as ( -2.36 ) V),

- ( k ) is Coulomb's constant [tex](\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),[/tex]

- |q| is the magnitude of the point charge we want to find, and

- ( r ) is the distance from the point charge to the point where the electric potential is being measured (given as [tex]\( 2.93 \times 10^{-3} \) m[/tex]).

Rearranging the equation to solve for |q|, we get:

[tex]\[ |q| = \frac{V \cdot r}{k} \[/tex]]

Substituting the given values, we have:

[tex]\[ |q| = \frac{-2.36 \times 2.93 \times 10^{-3}}{8.99 \times 10^9} \][/tex]

[tex]\[ |q| = \frac{-2.36 \times 2.93 \times 10^{-3}}{8.99 \times 10^9} \][/tex]

[tex]\[ |q| \approx 7.68 \times 10^{-12} \, \text{C} \][/tex]

Now, the electric potential is negative, indicating that the charge is negative. So, the point charge is a negative charge with a magnitude of [tex]\( 7.68 \times 10^{-12} \) C.[/tex]

Answer:

(a). The time is 26.67 sec.

(b). The distance traveled during this period is 1066.9 m.

Explanation:

Given that,

Speed = 80 m/s

Acceleration = 3 m/s

Initial velocity = 0

(a). We need to calculate the time

Using equation of motion

[tex]v = u+at[/tex]

[tex]t = \dfrac{v-u}{a}[/tex]

Put the value into the formula

[tex]t = \dfrac{80-0}{3}[/tex]

[tex]t =26.67\ sec[/tex]

The time is 26.67 sec.

(b). We need to calculate the distance traveled during this period

Using equation of motion

[tex]s = ut+\dfrac{1}{2}at^2[/tex]

[tex]s = 0+\dfrac{1}{2}\times3\times(26.67)^2[/tex]

[tex]s =1066.9\ m[/tex]

The distance traveled during this period is 1066.9 m.

Hence, This is the required solution.