you buy a gold crown at a flea market. when you get home, you use your physics knowledge to check whether the crown is pure gold. you hang it form a scale and find its weigh to be 7.84 N. them you weigh the crown when it is completely in water, now the scale reads 6.84 N. The density of gold is 19300 kgm^-3. density of water is 1000kgm^-3. what is the buoyant force on the crown?

a wave moves by you with a speed of 6.0 m/s . the distance from a crest of this wave to the next trough is 1.7 m. Frequency = wave speed / wavelength[tex]= 6.0 m/s / 3.4 m = 1.8 Hz.[/tex]

The frequency of a wave is the number of complete oscillations it makes per unit of time. In this case, we can use the equation [tex]f = v/λ,[/tex]  where f is frequency, v is the wave speed, and λ is the wavelength. The distance from a crest to the next trough is half of the wavelength, so we can calculate the wavelength as 2 x 1.7 m = 3.4 m. Plugging in the given values, we get a frequency of 1.8 Hz. This means that the wave completes 1.8 full oscillations per second as it travels past a stationary observer.

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The y-component of the angular momentum is 0.998 kg m^2/s.

To determine the angular momentum of the particle about the origin, we need to use the formula:

L = r x p

where L is the angular momentum, r is the position vector, and p is the momentum vector. In this case, we have:

m = 70 g = 0.07 kg
r = (4.6i - 6.2j) m
v = (3.1i - 8.5k) m/s
p = mv = (0.217i - 0.595k) kg m/s

To find the x-component of L, we need to take the cross product of the position vector with the momentum vector, and then take the x-component of the resulting vector:

L = r x p = | i    j    k |
            | 4.6 -6.2 0 |
            | 0.217 0 -0.595 |
   = (0)(-0.595 - 0) - (-6.2)(0.217 - 0) + (4.6)(0 - 0.217)i
   = -1.342i

Therefore, the x-component of the angular momentum is -1.342 kg m^2/s.

To find the y-component of L, we can take the y-component of the cross product:

L_y = r x p = | i    j    k |
               | 4.6 -6.2 0 |
               | 0.217 0 -0.595 |
    = (0.595)(0 - 0) - (0)(-0.217 - 0) + (4.6)(0.217 - 0)i
    = 0.998i

Therefore, the y-component of the angular momentum is 0.998 kg m^2/s.

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The power of the lens in diopters is P = 1/0.026 = 38.5 dioptres.

The power of the lens needed to magnify the reading material by a factor of 3.5x when a book is placed 9.0 cm behind a lens is 38.5 dioptres.

The focal length of the lens is given by the reciprocal of the magnification (M) multiplied by the distance of the object from the lens (d): f = 1/M x d.

In order to magnify the reading material by a factor of 3.5x when a book is placed 9.0 cm behind a lens, a diverging or converging lens is needed. The power of the lens in diopters can be calculated using the equation

P = 1/f, where P is the power in dioptres and f is the focal length in metres.

In this case, f = 1/3.5 x 0.09 = 0.026 m.

Thus, the power of the lens in diopters is P = 1/0.026 = 38.5 dioptres. Therefore, the power of the lens needed to magnify the reading material by a factor of 3.5x when a book is placed 9.0 cm behind a lens is 38.5 dioptres.

The type of image this would be is a magnified and real image. Magnified because the magnification factor is greater than one, and real image because it is formed on the same side of the lens as the object. It is also upright because the object is above the principal axis and the image below the principal axis. It is not inverted because the object and image appear the same way.  Lastly, it is not a virtual image because the image is formed by the refraction of light, not by the reflected rays.

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(a) immediately after the switch is closed:

IL = 0 A (inductor acts as an open circuit), I2 = 0 A (no current can flow through the circuit)

(b) a long time after the switch has been closed:

IL = 0 A (inductor acts as a short circuit), I2 = 10.0 mA (current flows only through R2)

(c) immediately after the switch is open:

IL = 5.89 mA, I2 = 0.0345 A (current flows through the inductor and R1)

(d) a time 4.712e-04 s after the switch is open:

IL = 1.94 mA, I2 = 0.0345 A (inductor current decreases and current still flows only through R1)

In (a), when the switch is closed after being open for a long time, the inductor acts as an open circuit and no current flows through it. Also, no current can flow through R2, so I2 = 0 A.

In (b), after the switch has been closed for a long time, the inductor acts as a short circuit and all the current flows through R1 and R2. Therefore, I2 = 10.0 mA.

In (c), when the switch is open, the inductor tries to maintain the current flowing through it and acts as a source. The current flows through R1 and the inductor, while no current flows through R2. Therefore, I2 = 0.0345 A.

In (d), the current in the inductor decreases exponentially because of the self-inductance, and some current flows through R1 and R2. Therefore, I2 remains constant at 0.0345 A, and IL decreases to 1.94 mA.

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It takes approximately 1.77 seconds for the cord to completely unwind from the wheel.

The angular acceleration of the wheel is given as 2 rad/s^2. We can use the kinematic equation for angular motion to determine the time it takes for the wheel to complete one revolution, which is the time it takes for the cord to unwind once from the wheel:

[tex]\theta = 2 * \pi\\\omega_i = 0\\\alpha = 2 rad/s^2\\[/tex]

t = ?

[tex]\theta = \omega_i * t + (1/2) * \alpha * t^2\\[/tex]

Solving for t:

[tex]t = \sqrt{(2 * \theta / \alpha)}\\t = \sqrt{(2 * \pi / 2 rad/s^2)}\\t = \sqrt{(\pi)\\[/tex]

t = 1.77 s (approx)

The rate at which the angular velocity of an item varies over time is known as its angular acceleration. In other words, it measures how quickly an object is changing its speed and/or direction of rotation. Angular acceleration is expressed in units of radians per second squared (rad/s²) or degrees per second squared (deg/s²).

When an object rotates, it experiences a torque that causes its angular acceleration. The amount of torque required to produce a given angular acceleration depends on the object's moment of inertia, which is a measure of how resistant an object is to changes in its rotation. Angular acceleration plays an important role in many physical systems, such as the motion of planets, the spinning of gyroscopes, and the movement of vehicles around corners.

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The wavelength of the matter wave associated with an electron in the n=11 electronic orbit in the hydrogen atom is 464 nm.

The wavelength of the matter wave associated with an electron in the n=11 electronic orbit in the hydrogen atom can be calculated using the de Broglie equation:

wavelength = h/p

where h is Planck's constant and p is the momentum of the electron. The momentum of the electron can be calculated using the formula:

p = mv

where m is the mass of the electron and v is its velocity.

Since we know that the electron is in the n=11 electronic orbit in the hydrogen atom, we can use the formula for calculating the radius of the electronic orbit:

r = n^2*(h^2/4*pi^2*m*e^2)

where n is the quantum number, m is the mass of the electron, and e is the charge of the electron.

Substituting the given values, we get:

r = 11^2*(h^2/4*pi^2*m*e^2)

Now, we can calculate the velocity of the electron using the formula:

v = sqrt((ke^2)/r)

where k is the Coulomb's constant and e is the charge of the electron.

Substituting the given values, we get:

v = sqrt((9*10^9*1.6*10^-19)^2/(11^2*(6.63*10^-34)^2*9.11*10^-31))

v = 1.57*10^6 m/s

Finally, we can calculate the momentum of the electron:

p = mv

p = (9.11*10^-31)*(1.57*10^6)

p = 1.43*10^-24 kg*m/s

Substituting the calculated momentum into the de Broglie equation, we get:

wavelength = h/p

wavelength = (6.63*10^-34)/(1.43*10^-24)

wavelength = 4.64*10^-10 m

Converting the wavelength from meters to nanometers, we get:

wavelength = 464 nm

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If there were friction present on the table, it would cause a resistive force that would oppose the motion of the mass on the table.

Friction results in a decrease in the overall speed and acceleration of the mass-pulley system. The friction force would also cause heat to be generated, which would lead to an increase in the system's temperature. Additionally, the presence of friction would affect the tension in the string connecting the mass to the pulley, which would have to work against the friction force to maintain the motion of the system.

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Number of bright fringes = 7

To calculate the number of bright fringes (visible maxima) in a diffraction pattern, we can use the formula for the diffraction grating equation:

n * λ = d * sinθ

Where n is the order of the bright fringe, λ is the wavelength of light (520 nm), d is the distance between adjacent slits in the grating, and θ is the angle of the diffracted light.

First, we need to find the distance between adjacent slits (d). The grating has 600 lines/mm, so we can calculate d as:

d = 1 / 600 lines/mm = 1/600 mm = 1.6667 * 10^(-6) m

Now, we can find the maximum order (n_max) that is visible using the grating equation:

n_max * 520 * 10^(-9) m = 1.6667 * 10^(-6) m * sin90°
n_max = 1.6667 * 10^(-6) m / 520 * 10^(-9) m
n_max ≈ 3.2

Since n must be an integer, the maximum order is n = 3.

To find the number of bright fringes, we count the orders on both sides of the central maximum (n = 0). So, there are 2 * 3 + 1 = 7 bright fringes in total.

The correct answer is b) 7.

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To solve this problem, we can use the conservation of angular momentum. The initial angular momentum (L_initial) is equal to the final angular momentum (L_final).

L_initial = L_final

The angular momentum is given by the product of the moment of inertia (I) and angular speed (w). So,

I0 * w0 = (1/3)I0 * w_final

Now, we can solve for w_final:

w_final = (I0 * w0) / ((1/3)I0)

w_final = 3 * w0

So, the answer is:

E.) 3w0

The correct answer is **D.) sqrt(3)w0**. This is because the law of conservation of angular momentum states that the initial angular momentum must be equal to the final angular momentum. The initial angular momentum is I0w0 and the final angular momentum is (1/3)I0 * w, where w is the final angular speed. Solving for w, we get w = (I0w0) / ((1/3)I0) = 3w0. Taking the square root of both sides, we get sqrt(w) = sqrt(3w0) or w = sqrt(3)w0.

The maximum current in the resistor is 5.31 A and the average power dissipated by the resistor is 338.03 W.

In this scenario, we have a step-down transformer with a turn ratio of 34:64 (secondary to primary). This means that the voltage in the secondary coil will be lower than the voltage in the primary coil.

Using the formula

Vp/Vs = Np/Ns, where Vp and Vs are the voltages in the primary and secondary coils, and Np and Ns are the numbers of turns in the primary and secondary coils, we can find the voltage in the secondary coil to be:

Vs = Vp(Ns/Np) = 120(34/64) = 63.75 V

To find the maximum current in the 12.0-Ω resistor connected to the secondary coil, we can use Ohm's law:

I = V/R = 63.75/12.0 = 5.31 A

For the average power dissipated by the resistor, we can use the formula P = VI, where V is the voltage across the resistor (which we just found to be 63.75 V) and I is the current through the resistor (which we just found to be 5.31 A):

P = VI = (63.75)(5.31) = 338.03 W

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If bulk is 3.6 kg lenght is 2.6 m turns around a direction passing toward the rod, rate 39 file(s. speed of 12 s actually the gravitational acceleration 99.7 J .

The definition of rotational energy, also referred angular kinetic energy, is as follows: It is the portion of an object's total kinetic energy that results from rotation. The angular velocity and the square of the angular velocity have a direct relationship with rotational kinetic energy.

The The International Systems of Units' (Symbol: J) unit of energy is the joule (/dul/ JEWEL, /dal/ JOWL) (SI). It is equivalent to the work generated with a force .

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A capacitor of approximately 3.14 μF in series with a 100-ohm resistor and a 23 mH inductor will give a resonance frequency of 900 Hz.

To calculate the capacitance required for resonance at 900 Hz, we can use the formula: resonance frequency = 1 / (2π√(L*C))
Where L is the inductance in henries and C is the capacitance in farads.

resonance frequency = 900 Hz
resistor = 100 Ω
inductance = 23 mH = 0.023 H

Substituting the values:

900 = 1 / (2π√(0.023*C))

Solving for C:

C = 1 / (4π² * 0.023 * 900²)
C = 0.0012 μF

Therefore, a capacitor of 0.0012 microfarads (or 1.2 nanofarads) in series with a 100 Ω resistor and a 23 mH inductor will give a resonance frequency of 900 Hz.
To find the value of the capacitor that will create a resonance frequency of 900 Hz when in series with a 100-ohm resistor and a 23 mH inductor, follow these steps:

Step 1: Understand the resonance frequency formula for an RLC series circuit:
f_resonance = 1 / (2π * √(L * C))

Step 2: Plug in the given values:
f_resonance = 900 Hz
L = 23 mH = 0.023 H (converting to henries)

Step 3: Rearrange the formula to solve for C:
C = 1 / (4π² * f_resonance² * L)

Step 4: Plug in the values and calculate the capacitance:
C = 1 / (4π² * (900)² * 0.023)
C ≈ 3.1405e-9 F

Step 5: Convert the capacitance to microfarads:
C = 3.1405e-9 F * (10^6 μF/F)
C ≈ 3.14 μF

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The form of induced voltage caused by wind or atmospheric conditions is typically referred to as "atmospheric static electricity." When the wind blows across an object or surface, it can create a separation of electric charges, which can result in an induced voltage.

This phenomenon is often observed during thunderstorms or other weather events when there is a buildup of static electricity in the atmosphere. The form of induced voltage caused by wind or atmospheric conditions is known as "electrostatic induction."

This phenomenon occurs when an electrically charged object affects the distribution of charges within a nearby neutral object, inducing a voltage in the neutral object without direct contact. In the case of wind and atmospheric conditions, the movement of air molecules can generate charge separation, creating a voltage difference that leads to electrostatic induction.

The form of induced voltage caused by wind or atmospheric conditions is known as electrostatic induction.

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a. When the switch is disconnected from point b and connected to point a, the capacitor will begin to charge. As a result, the voltage across the capacitor will increase exponentially with time. This is due to the charging process, which follows an exponential curve as the capacitor charges through the resistor.

When the switch is disconnected from point b and connected to point a, the capacitor will begin to charge. Initially, the capacitor has no charge, so it acts as a short circuit and allows the current to flow through the resistor.

As the capacitor charges, it starts to act as an open circuit, and the current flow decreases until it reaches zero when the capacitor is fully charged.

The voltage across the capacitor during this charging process increases exponentially with time, following the formula:
V(t) = [tex]V_0[/tex](1 - [tex]e^{(-t/RC)[/tex]),
where [tex]V_0[/tex] is the initial voltage,
t is the time elapsed since the switch was flipped,
R is the resistance of the resistor, and
C is the capacitance of the capacitor.

This exponential increase in voltage occurs because the rate of charging is proportional to the difference between the initial voltage and the maximum voltage that the capacitor can reach. As the capacitor charges and its voltage approaches the maximum voltage, the rate of charging decreases, resulting in the exponential curve.

Therefore, the voltage across the capacitor will increase exponentially with time when the switch is disconnected from point b and connected to point a.

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The collapse of the Broughton Bridge in 1831 was caused by the soldiers marching in lock step.

When the soldiers marched in unison, their footsteps created a rhythmic vibration that was transmitted to the bridge. This vibration matched the natural frequency of the bridge, causing the amplitude of the bouncing to increase gradually. As the amplitude increased, the bridge became unstable, and eventually, it collapsed.
After the collapse, soldiers were ordered to "break step" or march at their own individual rates while crossing bridges. This was to prevent the recurrence of such incidents. When soldiers march in lock step, they create a synchronized force that can be transmitted to the bridge, causing the amplitude of the vibrations to increase. However, when soldiers march at their own individual rates, the forces they create are not synchronized, and the bridge does not vibrate in a consistent pattern. This reduces the chances of the bridge collapsing due to the march of the soldiers.

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The magnitude of the magnetic field at the center of the loop is (A)1.2 x 10^-5 T if the wire carries a current of 6.0 A. The correct option is A.

To find the magnetic field at the center of the loop, we can use the formula for the magnetic field generated by a circular current loop:
B = (μ₀ * I) / (2 * π * r)
Where B is the magnetic field, μ₀ is the permeability of free space (4π x 10^-7 Tm/A), I is the current (6.0 A), and r is the radius of the loop (0.1 m).

1: Plug the values into the formula.
B = (4π x 10^-7 Tm/A * 6.0 A) / (2 * π * 0.1 m)
2: Simplify the expression.
B = (24π x 10^-7 Tm) / (0.2π m)
3: Cancel out the π terms and perform the calculation.
B = (24 x 10^-7 T) / 0.2
B = 1.2 x 10^-5 T
So, the magnitude of the magnetic field at the center of the loop is 1.2 x 10^-5 T. Therefore, option (A) is correct.

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a-the period of the oscillation is 1.00 s, b-the amplitude of the oscillation is 0.0420 m, c- the maximum speed of the mass is 0.264 m/s, d- the total energy of the system is 0.00807 J.

We can use the equations for simple harmonic motion to solve this problem.

a) The period (T) of the oscillation is given by:

T = 1/f

where f is the frequency of the oscillation.

Substituting f = 1.00 Hz, we get:

T = 1/1.00 Hz = 1.00 s

b) The amplitude (A) is the maximum displacement of the mass from its equilibrium position. We are given that at t = 0 s, the mass is at x = 4.20 cm. Therefore,

the amplitude is:

A = 4.20 cm = 0.0420 m

c) The maximum speed (v_max) of the mass occurs when it passes through the equilibrium position. At this point, the velocity of the mass is equal to the amplitude times the angular frequency (ω). The angular frequency (ω) is given by:

ω = 2πf

where f is the frequency of the oscillation.

Substituting f = 1.00 Hz, we get:

ω = 2π(1.00 Hz) = 6.28 rad/s

Therefore, the maximum speed is:

v_max = Aω = (0.0420 m)(6.28 rad/s) = 0.264 m/s

d) The total energy (E) of the system is the sum of the kinetic energy (K) and the potential energy (U). At the equilibrium position, all of the energy is potential energy. At the maximum displacement, all of the energy is kinetic energy. Therefore, the total energy is constant and equal to the potential energy at the equilibrium position:

E = U = (1/2)kA

where k is the spring constant.

To find k, we can use the formula for the frequency of simple harmonic motion:

f = 1/(2π) √(k/m)

where m is the mass attached to the spring.

Substituting m = 0.245 kg and f = 1.00 Hz, we get:

k = (2πf)^2m = (2π(1.00 Hz))(0.245 kg) = 9.56 N/m

Substituting this value for k and A = 0.0420 m, we get:

E = U = (1/2)(9.56 N/m)(0.0420 m)= 0.00807 J.

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The electric field has a magnitude of 3. 0 N/C at a distance of 60 cm from a point charge. The point charge has a charge of 1.0 x 10⁻⁹ C.

The equation: yields the electric field.

E = k×q/r²

where k is the Coulomb's constant, which has a value of 8.99 x 10⁹ N*m2/C2, and E is the electric field, q is the charge, r is the distance from the point charge.

At a distance of 60 cm from the point charge, the electric field has a magnitude of 3.0 N/C. We obtain r = 0.6 m by converting the distance to metres. When we enter the values, we obtain:

3.0 = (8.99 x 10⁹) ×q / (0.6)²

By calculating q, we obtain:

q = (3.0 × (0.6)²) / 8.99 x 10⁹

q = 1.0 x 10⁻⁹ C

As a result, the point charge has a charge of 1.0 x 10⁻⁹ C.

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a) The equation if 2D system point is : x' = y and y' = (-k/m)x - (b/m)y.

b) The fixed point at the origin is a center if (b² - 4mk) < 0, a stable node if (b² - 4mk) > 0 and b < 2sqrt(mk), and an unstable node if (b² - 4mk) > 0 and b > 2sqrt(mk).

a) Rewrite the damped harmonic oscillator equation as a 2D system point by introducing a new variable y = x'. Then the equation becomes a system of two first-order differential equations: x' = y and y' = (-k/m)x - (b/m)y.

b) In this case, since b = 0, the fixed point at the origin is a center. This means that the solutions to the system oscillate around the origin without converging to or diverging from it. The type of fixed point could change if the damping coefficient b is varied.

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For two resistors in series, in a series circuit, the voltage is divided among the components such that the higher resistance has the larger voltage and the component with the lower resistance has the smaller voltage. Therefore, the correct answer is c.

In a series circuit, the voltage is distributed among the components according to their resistance values.

Components with higher resistance will have a larger voltage drop across them, while components with lower resistance will have a smaller voltage drop.

This is in accordance with Ohm's Law (V = IR), where V is the voltage, I is current, and R is the resistance.

Since the current is the same in a series circuit, the voltage across a component is directly proportional to its resistance.

Therefore option "c" is the correct answer.

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A 101 g yo-yo swinging at 0.95 m/s has 0.046 Joules of kinetic energy.

Let's calculate the kinetic energy of the yo-yo using the terms mass, velocity, and the kinetic energy formula.

1. Mass (m) = 101 g (convert to kg) = 0.101 kg
2. Velocity (v) = 0.95 m/s
3. Kinetic energy (KE) formula: KE = 0.5 * m * v^2

Now, let's plug in the values:

KE = 0.5 * 0.101 kg * (0.95 m/s)^2

First, square the velocity:

0.95 m/s * 0.95 m/s = 0.9025 (m^2/s^2)

Next, multiply by mass and 0.5:

0.5 * 0.101 kg * 0.9025 (m^2/s^2) = 0.0456475 kg * (m^2/s^2)

The unit kg*(m^2/s^2) is the same as the unit Joule (J).

So, the yo-yo has 0.0456475 J of kinetic energy. To represent it with an appropriate number of significant figures, we can round it to 0.046 J (since the least number of significant figures in the given data is 2).

In summary, a 101 g yo-yo swinging at 0.95 m/s has 0.046 Joules of kinetic energy.

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Answer:

A cookie pan that's been heated by an oven will burn your hand because the pan is a solid object and has a much higher thermal conductivity than the air inside the oven. When the pan is heated in the oven, it absorbs the heat and becomes very hot. When you touch the hot pan, the heat energy flows from the pan to your hand, causing burns.

On the other hand, the air inside the oven has a lower thermal conductivity than the pan. This means that the air does not transfer heat as quickly as the pan does. Therefore, although the air inside the oven is hot, it does not burn your hand as easily as the hot pan would.

In other words, the pan is a better conductor of heat than the air inside the oven, which is why it can burn your hand even though they are at the same temperature.

Explanation:

The angular frequency (ω) of this spring/mass system, that oscillates up and down on a spring that has a force constant of 90 N/m, is approximately 3.888 rad/s.

To find the angular frequency of the spring/mass system, we need to use the following formula:

ω = √(k/m)

where ω is the angular frequency, k is the force constant (spring constant), and m is the mass of the object.

Given values:
- Mass (m) = 5.95 kg
- Force constant (k) = 90 N/m

Now, we can plug these values into the formula:

ω = √(90 N/m / 5.95 kg)

ω ≈ √(15.12605042)

ω ≈ 3.888 rad/s

So, the angular frequency (ω) of this spring/mass system is approximately 3.888 rad/s.

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Answer:

300 N

Explanation:

We can use the principle of Pascal's law to solve this problem. According to this law, pressure applied to a confined fluid is transmitted undiminished to every part of the fluid and to the walls of the container.

In this case, the fluid is confined in two cylinders with pistons of different sizes. The force applied on the larger piston is transmitted through the fluid and acts on the smaller piston, resulting in a larger force on the smaller piston.

We know that the force applied on the larger piston is 1500 N and the area of the larger piston is 25 m^2. Therefore, the pressure applied on the fluid is:

P = F/A = 1500 N / 25 m^2 = 60 Pa

This pressure is transmitted undiminished to every part of the fluid and acts on the smaller piston, which has an area of 5 m^2. Therefore, the force acting on the smaller piston is:

F = P × A = 60 Pa × 5 m^2 = 300 N

Therefore, a force of 300 N is needed to lift the crate.

The Conditions Young's double slit experiment uses two coherent sources of light placed at a small distance apart. Usually, only a few orders of magnitude greater than the wavelength of light are used. Young's double slit experiment helped in understanding the wave theory of light.

The double-slit experiment demonstrates that light and matter may exhibit both conventionally defined waves and particles; moreover, it demonstrates the inherently probabilistic nature of quantum mechanical events. Thomas Young initially performed this sort of experiment in 1801, as proof of visible light's wave behavior.

The Light was assumed to be made up of either waves or particles at the time. Around a hundred years later, at the dawn of modern physics, it was discovered that light could indeed exhibit wave-like and particle-like behavior.

If the light source is not coherent or monochromatic, the fringes will be blurred and there will be no interference pattern.

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A spring's spring constant is around 415.9 N/m, and when a spring is compressed by 7.9 cm from its relaxed length, it can store 21 j of elastic potential energy.

The following equation gives the elastic potential energy (U) held within a spring:

[tex]U = (1/2) * k * x^2[/tex]

where k is the spring constant and x is the displacement from the spring's relaxed length.

When the spring is compressed by 7.9 cm from its relaxed length, 21 J of elastic potential energy is stored in the spring. With this knowledge, we can construct the equation shown below:

[tex]21 J = (1/2) * k * (0.079 m)^2[/tex]

Solving for k, we get:

[tex]k = 21 J / [(1/2) * (0.079 m)^2] = 415.9 N/m[/tex]

The spring constant is a physical quantity that describes the relationship between the force exerted on a spring and the resulting displacement or deformation of the spring from its equilibrium position. It is a measure of the stiffness of the spring and is denoted by the letter k. The spring constant is a fundamental property of springs and is used extensively in physics and engineering applications.

The spring constant is defined as the ratio of the force applied to the spring to the resulting displacement of the spring. Mathematically, it is expressed as k = F/x, where F is the force applied to the spring and x is the resulting displacement of the spring from its equilibrium position.

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(a) To find the angular acceleration of the tires, we first need to find the initial and final angular velocities. The number of revolutions the tires make is not directly useful, so we need to convert it to an angle in radians.

The circumference of the tire is:

C = πd = π(0.90 m) = 2.83 m

So, each revolution covers an arc length of 2.83 m. Therefore, 67 revolutions correspond to an arc length of:

s = 67 × 2.83 m = 189.61 m

The initial velocity of the car is:

v1 = 85.0 km/h = 23.6 m/s

The final velocity of the car is:

v2 = 63.0 km/h = 17.5 m/s

The average velocity during the deceleration is:

v_avg = (v1 + v2) / 2 = (23.6 m/s + 17.5 m/s) / 2 = 20.55 m/s

The time it takes for the car to go from 85.0 km/h to 63.0 km/h is:

Δt = (v2 - v1) / a

where a is the angular acceleration. We can relate the linear acceleration a_l of the car to the angular acceleration a by:

a_l = a r

where r is the radius of the tire. In this case, r = 0.45 m.

The linear acceleration of the car is:

a_l = (v2 - v1) / Δt = (17.5 m/s - 23.6 m/s) / Δt = -6.10 m/s[tex]^2[/tex]

So, the angular acceleration of the tires is:

a = a_l / r = (-6.10 m/s[tex]^2[/tex] / 0.45 m = -13.56 rad/s^2 (note the negative sign indicates deceleration)

(b) If the car continues to decelerate at this rate, we can find the time it takes to stop using the equation:

v = v0 + at

where v is the final velocity (0 m/s), v0 is the initial velocity (23.6 m/s), a is the linear acceleration (-6.10 m/s^2), and t is the time we want to find.

Rearranging this equation, we get:

t = (v - v0) / a = (0 m/s - 23.6 m/s) / (-6.10 m/s[tex]^2[/tex] = 3.87 s

(c) To find the distance the car travels during this time, we can use the equation:

s = v0t + 1/2 at[tex]^2[/tex]

where s is the distance we want to find.

Plugging in the values we know, we get:

s = (23.6 m/s)(3.87 s) + 1/2 (-6.10 m/s[tex]^2)(3.87 s)^2[/tex] = 90.9 m

Therefore, the car travels a total distance of 189.61 m + 90.9 m = 280.51 m before coming to a stop.

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a. The magnetic latitude of the lavas erupted at 60°N, 51° magnetic latitude. b. The pole position was at 39.4°N, 144.4°E. The lavas indicate they were erupted on a continent that has moved from its original location.

c. A 5° error in the inclination angle measurement would result in an error of approximately ±9° in the calculated paleolatitude.

a. The angle of inclination is equal to the magnetic latitude, which can be found using the magnetic dipole equation. At 60°N with an inclination of 37°, the magnetic latitude is 51°.

b. The direction of magnetization and inclination angle provide information about the Earth's magnetic field at the time of eruption. The pole position can be calculated using the paleomagnetic data. The pole position was at 39.4°N, 144.4°E, which is significantly different from the current pole position. This indicates that the continent on which these lavas occur has moved from its original location.

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The ranking of wavelengths in order of increasing energy is as follows:

c. Infrared < b. Microwaves < a. X-Ray

The energy of electromagnetic radiation is directly proportional to its frequency and inversely proportional to its wavelength. This means that shorter wavelengths have higher energy, and longer wavelengths have lower energy.

Infrared radiation has longer wavelengths than microwaves and X-rays, so it has the lowest energy. Infrared radiation is commonly associated with heat, and is emitted by warm objects.

Microwaves have shorter wavelengths than infrared radiation, but longer wavelengths than X-rays. They are used in microwave ovens, telecommunications, and other applications.

X-rays have the shortest wavelengths and the highest energy of the three types of radiation. They are used in medical imaging and other applications that require high-energy radiation.

Therefore, the ranking of these wavelengths in order of increasing energy is: infrared, microwaves, and X-rays.

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(a) To produce a total charge of 4.55 x [tex]10^-^1^2[/tex] C, we need  2.84 x [tex]10^7[/tex] protons.

(b) The magnitude of the force between the gold nucleus and the electron is 7.31 x [tex]10^-^8[/tex] N.

(a) To calculate the number of protons needed to produce a total charge of 4.55 · 10-12 C, we can use the fact that the charge of a single proton is e=1.60 x [tex]10^-^1^9[/tex] C:

Number of protons = total charge / charge of one proton = 4.55 x [tex]10^-^1^2[/tex] C / 1.60 x [tex]10^-^1^9[/tex] C ≈ 2.84 x [tex]10^7[/tex] protons

(b) To calculate the magnitude of the force between a gold nucleus (+79e) and an electron on an orbit with radius 4.97·10-12 m, we can use Coulomb's law:

F = k * |q1| * |q2| / [tex]r^2[/tex]

where k is Coulomb's constant (k=8.99 x [tex]10^9[/tex] N·[tex]m^2[/tex]/[tex]C^2[/tex]), q1 and q2 are the charges of the particles, and r is the distance between them.

Here, we have q1=+79e and q2=-e (the charge of an electron), so we can simplify:

F = k * |79e| * |e| / (4.97 x [tex]10^-^1^2[/tex] m)² ≈ 7.31 x [tex]10^-^8[/tex] N

Therefore, the magnitude of the force between the gold nucleus and the electron is approximately 7.31 x [tex]10^-^8[/tex] N.

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