An Atwood machine is constructed using a disk of mass 2 kg and radius 24.8 cm. What is the acceleration of the system? The acceleration of gravity is 9.8 m/s 2 . Answer in units of m/s 2 .

To solve this problem we will apply the concepts related to Newton's second law, which defines force as the product between mass and acceleration. Mathematically this can be described as,

[tex]F = ma[/tex]

Here,

m = Mass

a = Acceleration

Taking as reference the mass of the second boxcar, the force applied would be

[tex]F = m_2 a[/tex]

[tex]F = (16300kg)(0.569m/s)[/tex]

[tex]F = 9274.7N[/tex]

Therefore the boxcars pull on each other with a force of 9274.7N

The boxcars pull on each other with equal and opposite forces, which can be calculated using the formula F = ma. The force that the boxcars pull on each other is 16521.1 N.

When the locomotive pulls the first boxcar, it exerts a force on it. According to Newton's third law of motion, the first boxcar exerts an equal and opposite force on the locomotive. Similarly, when the first boxcar pulls the second boxcar, it exerts a force on the second boxcar, and the second boxcar exerts an equal and opposite force on the first boxcar. Therefore, the boxcars pull on each other with equal and opposite forces.

To calculate the magnitude of this force, we can use the formula F = ma, where F is the force, m is the mass, and a is the acceleration. In this case, the mass of the first and second boxcars together is 12,700 kg + 16,300 kg = 29,000 kg. Plugging in the values, we get F = (29,000 kg)(0.569 m/s²).

The force that the boxcars pull on each other is 16521.1 N (rounded to four significant figures).

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Final answer:

For a step-up transformer with 22 primary turns and 800 secondary turns, to produce an output of 5300 V at 16 mA, the required input voltage is 145.75 volts and the input current needed is 0.5818 amperes.

Explanation:

Calculating Input Current and Voltage for a Step-Up Transformer

The student's question involves a step-up transformer with a known number of turns in the primary and secondary coils, a given secondary voltage, and a secondary current. To find the required input current and voltage, we can use the transformer equations that relate the primary and secondary sides of the transformer:

 Primary voltage (VP) / Secondary voltage (VS) = Number of turns in the primary coil (NP) / Number of turns in the secondary coil (NS)

 Primary current (IP) * Number of turns in the primary coil (NP) = Secondary current (IS) * Number of turns in the secondary coil (NS)

We're given:

 NP = 22 turns

 NS = 800 turns

 IS = 16 mA = 0.016 A

 VS = 5300 V

To find the input voltage VP:

VP = (NP / NS) * VS = (22 / 800) * 5300 V = 145.75 V

To find the input current IP:

IP = (NS / NP) * IS = (800 / 22) * 0.016 A = 0.5818 A

Therefore, the required input voltage is 145.75 volts, and the required input current is 0.5818 amperes.

Answer : The volume of the tank is, 1.54 mL

Explanation :

To calculate the volume of gas we are using ideal gas equation:

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

where,

P = pressure of gas = 300 kPa = 2.96 atm

Conversion used : (1 atm = 101.325 kPa)

V = volume of gas = ?

T = temperature of gas = [tex]77^oC=273+77=350K[/tex]

R = gas constant = 0.0821 L.atm/mole.K

w = mass of gas = 4.6 kg  = 4600 g

M = molar mass of air = 28.96 g/mole

Now put all the given values in the ideal gas equation, we get:

[tex](2.96atm)\times V=\frac{4600g}{28.96g/mole}\times (0.0821L.atm/mole.K)\times (350K)[/tex]

[tex]V=1541.98L=1.54mL[/tex]        (1 L = 1000 mL)

Therefore, the volume of the tank is, 1.54 mL

Answer: The heart pumps 124.2 billion cm³ of blood in a lifetime

Explanation:

as an adult the pulse rate average must be around 72 beats per minute.

The heart beats about 103,680 times in a day.

There are 365 days in a year

number of heart beat in a year = 365 days x 103,680 = 37,843,200 beats in a year

For every the heart pumps 50cm³ of blood,

Hence,

Amount of blood pump in a year = 50 x 37,843,200 = 1,892,160,000cm³ of blood pumped in a year.

Using the estimated lifespan average an individual is 69 years

So in a life time,

The human heart pumps = 1,892,160,000 x 69 years = 124,200,000,000

If the heart pumps 50cm³ of blood per beat, the heart pumps a total of 130,559,040,000 cm³ (130.6 billion cm³) of blood in a LIFETIME.

The human heart beats approximately 108,000 times per day, amounting to nearly 3 billion beats in a 75-year lifespan and pumps about 2.6 million gallons of blood. To estimate the volume in cubic meters, we use the flow rate of 5 L/min over a 75-year period and convert liters to cubic meters.

Estimating Human Heart Beat and Blood Volume Pumped Over a Lifetime

The vital importance of the heart is evident in its tireless work throughout a person's life. Assuming an average heart rate of 75 beats per minute, we can estimate that a human heart beats about 108,000 times in one day, which amounts to more than 39 million times in one year, and nearly 3 billion times during a 75-year lifespan. When it comes to the volume of blood pumped, with each contraction pumping approximately 70 mL of blood, the heart pumps roughly 5.25 liters of blood per minute. This translates to about 14,000 liters per day, and over a year, the heart would pump approximately 10,000,000 liters, or roughly 2.6 million gallons of blood through an extensive network of vessels.

Answer:

(a) 8 m/s

(b) 5 s

Explanation:

(a)

Using newton's equation of motion,

v² = u²+2gs ..................... Equation 1

Where v = final velocity, u = initial velocity, g = acceleration due to gravity at the surface of moon, s = height reached.

make u the subject of the equation,

u² = v²-2gs

u = √(v²-2gs)................ Equation 2

Note: As the stone is thrown up, v = 0 m/s, g is negative

Given: v = 0 m/s, s = 20 m, g = -1.6 m/s²

Substitute into equation 2

u = √(0-2×20×[-1.6])

u = √64

u = 8 m/s.

(b)

Using,

v = u+ gt

Where t = time of flight to reach the maximum height.

Make t the subject of the equation,

t = (v-u)/g................................... Equation 3

Given: v =  0 m/s, u = 8 m/s, g = - 1.6 m/s²

Substitute into equation 3

t = (0-8)/-1.6

t = -8/-1.6

t = 5 seconds.

Answer:

The height of the oil column above the tank top is 8 cm.

Explanation:

By applying Bernoulli's equation between point A and B as shown in the attached diagram

[tex]P_{atm}+\rho_{oil}g(h+0.12)=P_{atm}+\rho_{water}g(0.06+0.12)[/tex]

Here

               [tex]P_{atm}+\rho_{oil}g(h+0.12)=P_{atm}+\rho_{water}g(0.06+0.12)\\898\times (h+0.12)=998 \times (0.06+0.12)\\(h+0.12)=\frac{998}{898} \times (0.18)\\h+0.12=0.20\\h=0.20-0.12\\h=0.08m \approx 8 cm[/tex]

So the height of the oil column above the tank top is 8 cm.

Answer:

[tex]E = 6.38\times 10^6~N/C[/tex]

Explanation:

The question states that we can approximate the line as an infinite wire. In that case, the electric field can be found by Gauss' Law.

We should draw an imaginary cylindrical surface with an arbitrary height, h, around the wire. The radius of the cylinder should be equal to 65.1 cm.

Gauss' Law:

[tex]\int \vec{E}d\vec{a} = \frac{Q_{enc}}{\epsilon_0}[/tex]

The integral in the left-hand side is not to be taken, because we know the area of the cylinder. The enclosed charge in the right-hand side is equal to the charge of the portion of the wire inside the imaginary surface.

The charge density of the wire is

[tex]\lambda = \frac{Q}{L} = \frac{230 \times 10^{-3}}{1000} = 2.3 \times 10^{-4}[/tex]

The charge enclosed by the imaginary surface is

[tex]Q_{enc} = \lambda h = 2.3\times 10^{-4}h[/tex]

Finally, Gauss' Law yields

[tex]E2\pi rh = \frac{\lambda h}{\epsilon_0}\\E = \frac{\lambda}{2\pi \epsilon_0r} = \frac{2.3 \times 10^{-4}}{2\pi\epsilon_0(65.1\times 10^{-2})} = 6.38\times 10^6~N/C[/tex]

Answer:

His pitching speed is 38 m/s.

Explanation:

Hi there!

Please see the attached figure for a better understanding of the problem.

The position of the ball at any time t is given by the following vector:

r = (x0 + v0 · t, y0 + 1/2 · g · t²)

Where:

r = position vector of the ball at time t.

x0 = initial horizontal position.

v0 = initial horizontal velocity.

t = time.

y0 = initial vertical position.

g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive).

Let's place the origin of the frame of reference at the throwing point so that x0  and y0 = 0.

When the ball reaches the ground, its position vector will be r1 (see figure). Using the equation of the vertical component of the position vector, we can find the time at which the ball reaches the ground. At that time, the horizontal component of the position is 30 m and the vertical component is -3.0 m (see figure):

y = y0 + 1/2 · g · t²  (y0 = 0)

y = 1/2 · g · t²

-3.0 m = 1/2 · (-9.8 m/s²) · t²

-3.0 m / -4.9 m/s² = t²

t = 0.78 s

Now, knowing that at this time x = 30 m, we can find v0:

x = x0 + v0 · t  (x0 = 0)

x = v0 · t

30 m = v0 · 0.78 s

v0 = 30 m / 0.78 s

v0 = 38 m/s

His pitching speed is 38 m/s.

The pitching speed can be calculated using the horizontal distance and the height of the ledge. By considering the horizontal motion and the effects of gravity, the time taken can be determined. Substituting the given values into the appropriate equations, the pitching speed can be calculated as 38 m/s.

The pitching speed can be calculated using the horizontal distance and the height of the ledge. To find the initial velocity, we can use the equation:

Vox = d / t

where Vox is the horizontal component of velocity, d is the horizontal distance, and t is the time taken. Since the ball is thrown horizontally, the vertical component of velocity is zero, and the only force acting on the ball is gravity in the downward direction. Therefore, we can use the equation:

[tex]h = 0.5 * g * t^2[/tex]

where h is the height of the ledge, g is the acceleration due to gravity, and t is the time taken. By rearranging the equation, we can find the time taken:

t = sqrt(2 * h / g)

Substituting the values given in the question, we can calculate the pitching speed:

Vox = d / t = d * sqrt(g / (2 * h))

Using the values d = 30m, h = 3.0m, and g = 9.8m/s^2, we can find the pitching speed:

Vox = 30m * [tex]sqrt(9.8m/s^2[/tex]) = 38 m/s

One of the characteristics of the luminous gas clouds is that they do not have direct affectation by some type of external electric or magnetic fields.

In addition, we must bear in mind that color is a variable that is depending on the gas in the mixture. Therefore its relationship with spectroscopy allows us to deduce that scientists take advantage of the wavelength spectrum to know the type of composition of one of the clouds. The speed of a cloud is measured by determining the Doppler shift of its spectral lines. From wine's law, wavelength of light emitting from the object depends on temperature of object

Therefore the correct option is D

Astronomers analyze starlight to obtain various pieces of information about stars, including their temperature, composition, and motion. The correct answer is (d) all of the above.

(a) Temperature: By examining the spectrum of starlight, astronomers can analyze the distribution of wavelengths or colors present in the light. The temperature of a star affects the intensity and distribution of light at different wavelengths.

(b) Composition: The spectrum of starlight also provides information about the chemical composition of stars. Different elements and molecules in a star's atmosphere absorb or emit light at specific wavelengths, creating characteristic absorption or emission lines in the spectrum.

(c) Motion: Through the analysis of starlight, astronomers can also determine the motion of stars. By studying the Doppler effect on spectral lines, which causes a shift in wavelength due to the motion of a star toward or away from Earth, astronomers can measure a star's radial velocity.

Therefore, by analyzing starlight, astronomers can gather information about a star's temperature, composition, and motion, making option (d) all of the above the correct choice

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The flexural strength of the ZrO2 block is 7680 lb/in^2. The flexural modulus of the ZrO2 block is 33546.74 lb/in^2.

To calculate the flexural strength of the ZrO2 block, we need to use the formula:

Flexural strength = (3F * L) / (2 * b * h^2)

where F is the applied force, L is the distance between supports, b is the width of the block, and h is the thickness of the block. Substituting the given values, we have:

Flexural strength = (3 * 400 lb * 8 in.) / (2 * 0.50 in. * (0.25 in.)^2) = 7680 lb/in^2

To calculate the flexural modulus, we can use the formula:

Flexural modulus = (F * L^3) / (4 * b * h^3 * y)

where y is the deflection of the block. Substituting the given values, we have:

Flexural modulus = (400 lb * (4 in.)^3) / (4 * 0.50 in. * (0.25 in.)^3 * 0.037 in.) = 33546.74 lb/in^2

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Answer: the average speed of the rat from the information given above is 0.7m/s

Explanation:

position is given as

x(t) = pt² + qt

finding the diffencial of x(t) with respect to t, we have

d(x(t))/dt = 2pt + q

we substitute the p = 0.36m/s² and q= -1.10 m/s

d(x(t))/dt = 2(0.36)t + (-1.10)

so, at t= 1s

d(x(t))/dt = 2*(0.36)*1 - 1.1 = 0.72 - 1.1 = -0.38m/s

at t= 4s

d(x(t))/dt = 2*(0.36)*4 - 1.10 = 2.88 - 1.10 = 1.78 m/s

To find the average speed,

average speed = (V1 + V2)/ 2

average speed = (1.78 + (-0.38))/2 = 0.7m/s

The speed is defined as the distance per unit of time. The unit of speed is m/s. The speed is a scalar quantity which means it only depends on the magnitude.

According to the question, The average speed of the mouse is 0.7m/s

The solution of the question is as follows:-

The required equation is:-

[tex]x(t) = pt^2 + qt[/tex]

The Finding the differential of x(t) with respect to t, we have

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

Put the value p = 0.36m/s² and q= -1.10 m/s

[tex]\frac{d(x(t)}{dt} = 2(0.36)t + (-1.10)[/tex]

so, at t= 1s

After solving it [tex]2*(0.36)*1 - 1.1 = 0.72 - 1.1 = -0.38m/s[/tex]

so,at t= 4s

After solving it =[tex]2*(0.36)*4 - 1.10 = 2.88 - 1.10 = 1.78 m/s[/tex]

The formula of average speed = [tex]\frac{(V1 + V2)}{2}[/tex]

[tex]= \frac{(1.78 + (-0.38))}{2} = 0.7m/s[/tex]

Hence, the average speed is 0.7m/s

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Answer:

The star will have a transverse speed of 315950.9 AU/year

Explanation:

d = 1/p

d = distance to star, measured in parsecs

p = parallax, measured in arcseconds = 0.5 arcsec/year

So, d = 1/0.5 = 2 parce

1 parsec = 3.26 light years

2 parce = 6.52 light years

⇒Transverse speed in AU / year = Distance/parallax

distance = 10pc = 2060000 AU

Transverse speed in AU / year =  2060000 Au/6.52 light years

Transverse speed  = 315950.9 AU/year

Therefore, A star (not Barnard's star) at a distance of 10 pc  observed to have a proper motion of 0.5 arcsec / year. Will have a transverse speed of 315950.9 AU/year

Answer:

The coyote crashes 66 m from the base of the cliff.

Explanation:

Hi there!

The equation of the position vector of the Coyote is the following:

r = (x0 + v0 · t + 1/2 · a · t², y0 + 1/2 · g · t²)

Where:

r = postion vector of the Coyote at time t.

x0 = initial horizontal position.

v0 = initial horizontal velocity.

t = time.

a = horizontal acceleration.

y0 = initial vertical position.

g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive).

Let's place the origin of the frame of reference at the edge of the cliff so that x0 and y0 = 0.

When the Coyote reaches the ground, the vertical component of its position vector (r1 in the figure) will be equal to -29 m. When the vertical component of the position vector is -29 m, the horizontal component will be equal to the horizontal distance traveled by the Coyote (r1x in the figure). So, let's find the time at which the y-component of the position vector is -29 m:

y = y0 + 1/2 · g · t²  (y0 = 0)

-29 m = -1/2 · 9.8 m/s² · t²

t² = -29 m / -4.9 m/s²

t = 2.4 s

Now, let's find the x-component of the vector r1 in the figure:

x = x0 + v0 · t + 1/2 · a · t²     (x0 = 0)

x = 24 m/s · 2.4 s + 1/2 · 3 m/s² · (2.4 s)²  

x = 66 m

The coyote crashes 66 m from the base of the cliff.

The problem involves determining the distance Wile E. Coyote travels horizontally before he crashes into the ground. It involves the use of motion equations to first calculate the time it takes for the coyote to hit the ground and then the distance it travels horizontally. The calculated distance is roughly 65.4 meters.

In this problem, we are asked to calculate the distance that Wile E. Coyote travels along the ground before he crashes.

To solve this, we need to use the laws of motion. First, since the coyote falls with a constant acceleration due to gravity, we can use the equation of motion to determine the time it takes for him to hit the ground:

29 = 0.5 * g * t2

By plugging in the acceleration due to gravity (g=9.8m/s2), we determine that t is approximately 2.44 seconds.

Next, we have to calculate how far the coyote travels horizontally. He starts with a velocity of 24m/s, but he also accelerates due to the rocket. So the distance he travels is given by:

x = v0t + 0.5*a*t2

Substituting the known values (v0 = 24m/s, a = 3m/s2, t = 2.44s), we find that the coyote travels roughly 65.4 meters along the ground before he crashes into it.

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To solve this problem we will apply the concepts related to the cinematic equations of angular motion. On these equations, angular acceleration is defined as the squared difference of angular velocity over twice the radial displacement. This is mathematically:

[tex]\alpha = \frac{\omega^2-\omega_0^2}{2\theta}[/tex]

Our values are,

[tex]\text{Initial angular velocity} = \omega_0 =6.36 rad/s[/tex]

[tex]\text{Final angular velocity} = \omega =0[/tex]

[tex]\text{Angular displacement} = \theta = 14.7rev = 29.4\pi rad[/tex]

Replacing,

[tex]\alpha = \frac{- 6.36^2}{29.4\pi}[/tex]

[tex]\alpha = -0.43rad/s^2[/tex]

Therefore the angular acceleration is [tex]-0.43rad/s^2[/tex]

The correct options are a) a. false b) b. false c) a.True d) a.True e) a.True f) a.True g) b. false.

(a) The given statement is incorrect, An electric field is the gradient of the potential. when the Potential is constant The electric field is zero. The correct option is b. false

(b)  If an electric field of space is zero, it does not imply that there is no charge . From Gauss law in its divergence form, It implies there are an equal number of opposite charges present. No statement can be said about the charges present in that region. The correct option is b. false.

(c) The statement is correct because the Intensity of an electric field is line integral of the electric potential. The correct otption is a. True

(d) Electric field lines always point from high potential to low potential. The correct option is a. True

(e) When the electric field strength is not zero, an electric potential is zero at all points on the equatorial line of the electric dipole. The correct option is a. True

(f) The statement is correct as conductors allow the free flow of charge within themselves. The correct statement is a. True

(g) It is incorrect as dielectric breakdown occurs when a charge buildup exceeds the electrical limit or dielectric strength of a material. The correct option is b. False

The options are a) false b) false c) True d) True e) True f) True g) false.

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Final answer:

The relationship between electric fields and electric potential is complex. Zero electric field does not guarantee zero electric potential, and equipotential surfaces arise when there is no change in potential, forming naturally around conductors in electrostatic conditions.

Explanation:

Addressing the statements related to electric fields and electric potential:

False: If the electric field is zero in some region of space, it does not imply that the electric potential is also zero; the potential could have a nonzero constant value.

True: If the electric potential is uniformly zero, the electric field must also be zero since a nonzero field would indicate a change in potential.

False: Zero electric potential at a point does not necessarily mean that the electric field is zero at that point.

True: Electric field lines always point from regions of higher to lower potential.

True: The value of the electric potential can be chosen to be zero at any convenient reference point.

True: In electrostatics, the surface of a conductor is an equipotential surface because the electric field within a conductor must be zero.

False: The breakdown voltage of air varies depending on conditions such as air density and humidity, and is generally accepted to be approximately 3×10⁶ V/m but the exact value can differ.

To solve this problem we will apply the concept of Newton's second law with which we will obtain the strength of the proton. We know the mass and the acceleration is given in the statement. Subsequently said Force by equilibrium can be matched the electrostatic force of Coulomb, defined as the product between the charge and the electric field. Our values are

[tex]m = 1.67*10^{-27}kg[/tex]

[tex]a = 10*10^{11} m/s^2[/tex]

Applying the Newton's second law,

[tex]F = ma[/tex]

[tex]F = (1.67*10^{-27}kg)( 10*10^{11} m/s^2)[/tex]

[tex]F = 1.67*10^{-15}N[/tex]

By the Coulomb's equation for electrostatic Force we have that

[tex]F = qE[/tex]

Remember that the charge of a proton is [tex]1.6*10^{-19}C[/tex]

Replacing we have,

[tex]1.67*10^{-15} = 1.6*10^{-19} E[/tex]

[tex]E = 10437.5 N/C[/tex]

Therefore the magnitude of the electric field at the proton's location is [tex]10437.5 N/C[/tex]

The magnitude of the electric field at the proton's location is 10,437.5 N/C.

The given parameters:

The magnitude of the electric field at the proton's location is calculated as follows;

F = ma

F = qE

qE = ma

[tex]E = \frac{ma}{q} \\\\E = \frac{1.67 \times 10^{-27} \times 10\times 10^{11}}{1.6 \times 10^{-19}} \\\\E = 10,437.5 \ N/C[/tex]

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Final answer:

The diameter of the iron ball bearing, which weighs 21.91 grams and is composed of iron atoms organized in a BCC structure, is roughly 1.62 cm.

Explanation:

To derive the diameter of a spherical iron ball bearing weighing 21.91 grams, given that iron atoms have an effective radius of 0.124 nm and are arranged in a Body-Centered Cubic (BCC) structure at room temperature, we need to calculate the volume of the iron ball and then find the diameter using the volume of a sphere formula. First, we will use the density of iron (7.9 g/cm³) to find the volume of the ball bearing:

V = mass / density = 21.91 g / 7.9 g/cm³ = 2.77342 cm³

Next, we use the volume of a sphere formula V = (4/3)πr³, where V is the volume and r is the radius, to find the diameter (d = 2r):

r³ = V / ((4/3)π) = 2.77342 cm³ / ((4/3)π) ≈ 0.52733 cm³

r ≈ 0.8092 cm

d = 2 * r ≈ 2 * 0.8092 cm ≈ 1.6184 cm

Therefore, the estimated diameter of the iron ball bearing is approximately 1.62 cm.

Answer:

The new position is 0.1865 m

Explanation:

As the context of the data is not available, thus following data is utilized from the question as attached above

x_relax=0.32 m

x_stiff=0.13 m

spring stiffness k=9 N/m

mass of block =0.073 kg

t=0.07 s

Velocity of the block is to be estimated thus

Force due to compression in spring is given as

F_s=k Δx

F_s=9(0.32-0.13)

F_s=1.71 N

Force on the block is given as

F_m=mg

F_m=0.073 x 9.8

F_m=0.71 N

Net Force

F=F_s-F_m

F=1.71-0.71 N

F=1 N

As Ft=Δp

So

Δp=1x0.07=0.07 kgm/s

Δp=p_final-p_initial

0.07=p_final-0

p_final=0.07 kgm/s

p_final=m*v_f

v_f=(p_final)/(m)

v_f=0.07/0.073

v_f=0.95 m/s

So now the velocity of the block is 0.95 m/s

time is 0.07 s

y_new=y_initial+y_travel

y_new=0.12+(0.95 x 0.07)

y_new=0.12+0.065

y_new=0.1865 m

So the new position is 0.1865 m

Final answer:

The new position of the bottom of the block at time t = 0.07 seconds is 1 m.

Explanation:

To find the new position of the bottom of the block at time t = 0.07 seconds, we can use the concept of average velocity. The average velocity is given by the change in position divided by the change in time. In this case, if we approximate the average velocity in the first time interval by the final velocity, we can say that the change in position is equal to the average velocity multiplied by the change in time. The new position can then be calculated by adding this change in position to the initial position.

Given that the initial position of the bottom of the block is at y = 0.12 m and the final velocity is approximated to be y = 1 m, we can calculate the change in position as:

Change in position = (Final velocity - Average velocity) * Change in time = (1 m - 0.12 m) * (0.07 s - 0 s) = 0.88 m

Therefore, the new position of the bottom of the block at time t = 0.07 seconds is y = 0.12 m + 0.88 m = 1 m.

Answer:

v = -4.22 x 10⁻⁴ m/s

Explanation:

given,

measured wavelength = 497.15 nm

Normally wavelength = 497.22 nm

Change in wavelength

Δ λ = 497.15 - 497.22

Δ λ = -0.07 nm

using Doppler's equation

[tex]\dfrac{\Delta \lambda}{\lambda}=\dfrac{v}{c}[/tex]

v is the speed of the star

c is the speed of light

[tex]\dfrac{-0.07\ nm}{497.22\ nm}=\dfrac{v}{3\times 10^8}[/tex]

      v = -4.22 x 10⁻⁴ m/s

Speed of the star moving is equal to v = -4.22 x 10⁻⁴ m/s

The speed of the star with respect to the Earth is -4.22× 10⁻⁴ m/s m/s. The negative sign indicates the star is moving away.

Given:

Observed wavelength (λ) = 497.15 nm

Rest wavelength (λ₀) = 497.22 nm

To calculate the speed of a star with respect to Earth. Doppler effect technique can be used.  The Doppler effect gives the relation between wavelength, speed, and speed of light.

The formula for the Doppler shift is given as:

Δλ / λ₀ = v / c

Δλ = λ - λ₀

Δλ = (497.15 x 10⁻⁹ m) - (497.22 x 10⁻⁹ m)

Δλ = -0.07 x 10⁻⁹ m

The speed of the star is evaluated as:

v = (Δλ / λ₀) x c

v = (-0.07 x 10⁻⁹ m / 497.22 x 10⁻⁹ m) x 299,792,458 m/s

v = -4.22× 10⁻⁴ m/s

Hence, the speed of the star with respect to the Earth is -4.22× 10⁻⁴ m/s m/s. The negative sign indicates the star is moving away.

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Answer:

17.97m/s

Explanation:

Density of air (ρ)air=1.23 kg/m3, and

Air speed (V) =20 m/sec, pressure gradient along the streamline, ∂p/∂x = 100N/m^3.

The equation of motion along the stream line directions:

considering the momentum balance along the streamline.

γsinθ-∂p/∂x=ρV(∂V/∂x)

Neglecting the effect of gravity , then γ=ρg=0

So, ∂p/∂x= -ρV(∂V/∂x)

∂V/∂x= - 100/(20X1.23)= -4.0650407/S

Also δV/δx=∂V/∂x

∂V/∂x=-4.0650407/S and δx=0.5 m

δV = (-4.0650407/S) *(0.5m)

δV = -2.0325203 m/S

So net air speed will be V+δV= -2.0325203+20 ≅17.96748 m/s

Approximately, V+δV=17.97m/s.

Using Bernoulli's equation and the given pressure gradient, we can calculate the airspeed at a point 0.5 m further along the streamline to be approximately 45.83 m/s.

To solve this problem, we can use Bernoulli's equation, which is a principle in fluid dynamics that states the total mechanical energy in a fluid system is constant if no energy is added or removed by work or heat transfer. It is formulated as p1 + 1/2 ρ v1² + ρgh1 = p2 + 1/2 ρ v2² + ρgh2.

In this case, we assume potential energy (ρgh) terms to be zero because there is no change in height, and the fluid (air) is incompressible. We are also given that the pressure gradient is 100 N/m³ which is effectively the change in pressure (Δp = p2 - p1), and the change in the distance along the streamline Δs = 0.5 m.

So we are left with: Δp + 1/2 ρ v1² = 1/2 ρ v2². Since ρ also cancels out, we are left with v2² = v1² + 2 Δp/ρ Δs. Plugging in given values we get v2 = √( 20² + 2*100*0.5) = √2100 = 45.83 m/s, assuming air density (ρ) is approximately 1.225 kg/m³ at sea level and at 15 °C.

Therefore, the airspeed at a point 0.5 m further along the streamline is approximately 45.83 m/s.

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Answer:

4

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

[tex]m_1[/tex] = Mass of Earth

[tex]m_2[/tex] = Mass of Moon

r = Distance between Earth and Moon

Old gravitational force

[tex]F_o=\dfrac{Gm_1m_2}{r^2}[/tex]

New gravitational force

[tex]F_n=\dfrac{Gm_1m_2}{(\dfrac{1}{2}r)^2}[/tex]

Dividing the equations

[tex]\dfrac{F_n}{F_o}=\dfrac{\dfrac{Gm_1m_2}{(\dfrac{1}{2}r)^2}}{\dfrac{Gm_1m_2}{r^2}}\\\Rightarrow \dfrac{F_n}{F_o}=\dfrac{\dfrac{Gm_1m_2}{\dfrac{1}{4}r^2}}{\dfrac{Gm_1m_2}{r^2}}\\\Rightarrow \dfrac{F_n}{F_o}=4[/tex]

The ratio is [tex]\dfrac{F_n}{F_o}=4[/tex]

The new force would be 4 times the old force

If the distance between the Earth and Moon were halved, the force of gravity between them would be quadrupled.

According to Newton's law of universal gravitation, the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this relationship is expressed as:

F= G*m*M/r^2

If the distance between the Earth and Moon were halved, the force of gravity between them would be quadrupled. This is because the force of gravity is inversely proportional to the square of the distance between two objects. So, if the distance is divided by 2, the new force would be multiplied by (2^2) = 4.

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Answer:

800

Explanation:

Focal length of telescope, F = 200cm

Focal length of eyepiece, f = 0.25

The magnifying power of a telescope is given as the ratio of the focal length of the objective of the telescope to the focal length of the lens. Mathematically:

M = F/f

Therefore, when F = 200cm and f = 0.25cm:

M = 200/0.25

M = 800

Answer:

-48cm

Explanation:

the following data are given

focal length of telescope=200cm,

focal length of the eyepiece=0.25cm

From the genera formula used to find the magnifying power which is expressed as

[tex]M=-\frac{fx_{o}}{f_{e}}[1+\frac{f_{e}}{d}][/tex]

where

[tex]f_{e} = focal length of thr eye piece\\ f_{o} =focal length of the telescope\\[/tex]

and d=least distance of distinct vision=25cm

if we substitute values into the formula, we arrive at

[tex]M=-\frac{fx_{o}}{f_{e}}[1+\frac{f_{e}}{d}]\\M=-\frac{200cm}{0.25cm}[1+\frac{0.25cm}{25cm}]\\M=-800cm[1+0.01]\\M=-800cm(1.01)\\M=-808cm \\M=-808cm[/tex]

hence from the answer, we can conclude that the magnifying power of the telescope is -808cm

Answer:

a. The apparent depth of the object is less than the real depth.

Explanation:

When we observe for any object lying at the bottom of the pool from the edge of the pool then we are actually viewing an object from an optically rarer medium into an optically denser medium.

The schematic shows the apparent view of the object due to the bending of the rays coming form the object to our eyes.

The rays when coming from a denser medium to a rarer medium they bend away from the normal of the interface.

The correct option is a. The apparent depth of the object is less than the real depth because of the refraction of light at the water-air interface.

Light bends away from the normal as it exits the water, making the object appear shallower than it actually is. When light travels from water to air, it bends away from the normal because water is denser than air.

This bending makes the object appear to be at a shallower depth than it actually is. The apparent depth of the object is therefore less than the real depth of the object.

To summarize, the correct statement is: The apparent depth of the object is less than the real depth. Option a is correct.

Answer:

D) result of the calculation is zero

Explanation:

Coulomb's Law is valid for only point-like particles. Since the ring is not a point-like, then we have to choose an infinitesimal portion (ds) of the ring, apply the Coulomb's Law to this portion and then integrate over the ring to find the total force.

The small portion (dq) will have the same charge density as the ring itself. Furthermore, the length of the infinitesimal portion is equal to the radius times the corresponding angle, dθ.

[tex]\lambda = \frac{Q}{2\pi R} = \frac{dq}{Rd\theta}\\dq = \frac{Qd\theta}{2\pi}[/tex]

Therefore, the force between the charge at the center and the small portion is

[tex]dF = \frac{1}{4\pi\epsilon_0}\frac{qdq}{R^2} = \frac{1}{4\pi\epsilon_0}\frac{qQd\theta}{2\pi R^2}[/tex]

Since force is a vector, we have separate its x- and y-components,

[tex]dF_x = \frac{1}{4\pi\epsilon_0}\frac{qQd\theta}{2\pi R^2}\cos(\theta)\\dF_y = \frac{1}{4\pi\epsilon_0}\frac{qQd\theta}{2\pi R^2}\sin(\theta)[/tex]

Now, we can integrate both of them over the ring.

[tex]F_x = \int\limits^{2\pi}_0 dF_x = \frac{1}{4\pi\epsilon_0}\frac{qQ}{2\pi R^2}\int\limits^{2\pi}_0\cos(\theta)d\theta = 0\\F_y = \int\limits^{2\pi}_0 dF_y = \frac{1}{4\pi\epsilon_0}\frac{qQ}{2\pi R^2}\int\limits^{2\pi}_0\sin(\theta)d\theta = 0[/tex]

Since the integration from 0 to 2π for sine and cosine functions results as zero.

Therefore, the force on the charge at the center of a uniformly distributed ring is equal to zero.

The total force exerted on a charge q placed at the center of a circle with a uniformly distributed charge Q along the circumference is zero due to the symmetry of the charge distribution.

When a charge Q is spread uniformly along the circumference of a circle with radius R, and a point particle with charge q is placed at the center of this circle, we must apply Coulomb's law to calculate the force exerted on the charge q. Thanks to the symmetry of the charge distribution, the forces exerted by individual segments of the charged circumference on the central charge will cancel each other out in every direction. Hence, while the distance from the charge q to any point on the circle is R, the resulting total force on charge q will be zero due to symmetry.

It's important not to confuse the circumference with other distances, such as the diameter (2R) or the circumferential length (2πR), as these are not relevant for calculating the force on the central charge in this symmetric setup. Therefore, the correct answer is that the result of the calculation is zero (Option D), because the uniform distribution of charge Q around the circle results in an equilibrium of forces.

Answer:

[tex]v=1.92m/s[/tex]

Explanation:

Given data

Length h=2.0m

Angle α=25°

To find

Speed of bob

Solution

From conservation of energy we know that:

[tex]P.E=K.E\\mgh=(1/2)mv^{2}\\ gh=(1/2)v^{2}\\v^{2}=\frac{gh}{0.5}\\ v=\sqrt{\frac{gh}{0.5}}\\ v=\sqrt{\frac{(9.8m/s^{2} )(2.0-2.0Cos(25^{o} ))}{0.5}}\\v=1.92m/s[/tex]

Given values:

As we know,

The conservation of energy:

→ [tex]Potential \ energy = Kinetic \ energy[/tex]

or,

→ [tex]mgh = \frac{1}{2} mv^2[/tex]

or,

→    [tex]v^2 = \frac{gh}{0.5}[/tex]

          [tex]= \sqrt{\frac{gh}{0.5} }[/tex]

By substituting the values, we get

          [tex]= \sqrt{\frac{(9.8)(2.0-2.0 Cos (25^{\circ}))}{0.5} }[/tex]

          [tex]= 1.92 \ m/s[/tex]

Thus the above answer is right.

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Answer:

Q_out,shell = 9Q

Explanation:

Given:

- Q_in,shell = -10 Q

- Q_sphere = +1Q

Find:

How much charge is on the outer surface?

Solution:

The electric field in the material of both the sphere and shell must be zero. The only way for this to occur is if the charge inside the

inner surface of the shell is such that its charge plus the solid's charge is zero. The rest of the excess charge from the shell moves to  the outside of the shell.

Hence,

                     Q_out,shell + Q_in,shell + Q_sphere = 0

                     Q_out,shell -10 Q + 1 Q = 0

                    Q_out,shell = 9Q

Final answer:

The charge on the outer surface of the hollow conducting sphere, which initially had a charge of -10Q and contains an inner sphere with a charge of +1Q, would be -9Q, as determined by Gauss' Law and the conservation of charge.

Explanation:

The student is asking about the charge distribution on concentric conducting spheres when one sphere is placed inside another and they each have different charges. According to Gauss' Law, when a charge is placed inside a conducting shell, it induces an equal and opposite charge on the inner surface of the shell to maintain an electric field of zero inside the material of the conductor. In the given scenario, the inner solid sphere has a charge of +1Q and the outer hollow sphere has a charge of -10Q.

By Gauss' Law, since the electric field inside a conductor must be zero, we know that the inner surface of the hollow outer sphere must have a charge of -1Q to cancel out the electric field from the +1Q charge of the inner solid sphere.

Considering charge conservation, if the outer sphere initially had a total charge of -10Q and now there is -1Q on the inner surface, the outer surface of the hollow sphere must have the remainder, which is -10Q + 1Q = -9Q. Therefore, the charge on the outer surface of the outer hollow sphere is -9Q.

Answer

given,

storage coefficient, S = 6.8 x 10⁻⁴

thickness of aquifer, t = 50 m

porosity of the aquifer, n = 25 % = 0.25

Density of the water, γ = 9810 N/m³

Compressibilty  of water,β = 4.673 x 10⁻¹⁰ m²/N

We know,

   S = γ t(nβ + α)

where, α is the compressibility of the aquifer

   6.8 x 10⁻⁴  =9810 x 50 x (0.25 x 4.673 x 10⁻¹⁰+ α)

     α = 1.269 x 10⁻⁹ m²/N

Expansability of water

            = n t β γ

            = 0.25 x 50 x 4.673 x 10⁻¹⁰ x 9810

            = 5.73 x 10⁻⁵

Answer:

E along –x; F along +x

Explanation:

The electric field at point P due to the negative point charge is directed along the -x axis, and the force experienced by the electron at point P due to this field is also along the -x axis.

The electric field at point P due to the negative point charge at x = 0 is directed along the -x axis. This is because electric field lines always point away from positive charges and toward negative charges. Since the negative charge is located at x = 0, the electric field at point P, which is on the positive x-axis, points in the opposite direction, i.e., along the -x axis.

The force experienced by the electron at point P due to this electric field will be in the same direction as the electric field, i.e., along the -x axis. Like charges repel each other, so the negative point charge will exert a repulsive force on the electron.

Using the concept of Newton's second law, as the elevator starts from rest and travels upward, the scale shows a value of 94.04 kg

Acceleration is the time derivative of velocity. Therefore acceleration can be given by;

[tex]a=a(t)=\frac{dv(t)}{dt} = \frac{d}{dt} (3t+0.2t^2) = 3+0.4t[/tex]

When t = 3s;

[tex]a= 3+(0.4\times 4)=4.6\,m/s^2[/tex]

As the elevator is moving upward the net force on the weighing scale is given using Newton's second law of motion;

[tex]F_{net}=m(a+g) =(64\,kg)\times (9.8\,m/s^2 +4.6\,m/s^2)=921.6\,N[/tex]

A weighing scale is usually caliberated to show the value in kilograms.

Therefore the weighing scale will show the reading;

[tex]m=\frac{F_{net}}{g} =\frac{921.6\,N}{9.8\,m/s^2}= 94.04\,kg[/tex]

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The apparent weight you would experience in the accelerating elevator is 925.12N. This higher reading is due to the addition of the upward acceleration of the elevator to the standard pull of gravity.

To solve this question, we first need to find the elevator's acceleration. The acceleration is the derivative of the velocity: v(t)=(3.0m/s²)t+(0.20m/s³)t² with respect to time t. The derivative is a(t) = 3.0m/s² +2×(0.20m/s³)×t.

Substitute t=4.0s into a(t) to get a(4.0s) = 3.0m/s² + 2×0.2×4.0 = 4.6m/s². This is the acceleration at t=4.0s.

The apparent weight, or reading on the scale, is given by the equation: F=m(g+a), where m is the mass (64 kg), g is the acceleration due to gravity (9.8 m/s²), and a is the acceleration of the elevator. Thus the apparent weight is F=64(9.8+4.6)=925.12 N.

This is your weight when the elevator is accelerating upward; you would feel heavier due to the addition of the elevator's upward acceleration to the natural gravitational pull.

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