20 POINTS
In order to maximize the acceleration of an object, one should
maximize the mass
maximize the force
minimize the velocity
maximize the inertia

Answer:

1 km

Explanation:

displacement =velocity ×time

displacement =40m/s ×25s

displacement =1000m equivalent to 1km

Answer:

Second Option

Explanation:

The "hard drive" or the second option is one of the main components of storing information on a computer. You already have a hard drive built into your computer, or laptop when you buy it, and you can buy additional hard drives in the form of plugins that can store even more data if your original hard drive becomes full of data.

Hope this helps.

Answer:

B

Explanation:

Whereas memory refers to the location of short-term data, storage is the component of your computer that allows you to store and access data on a long-term basis. Usually, storage comes in the form of a solid-state drive or a hard drive.

The x and y component of the ball's final velocity are respectively 7.35 m/s and  4.90 m/s.

The rate at which a body's displacement changes in relation to time is known as its velocity. Velocity is a vector quantity with both magnitude and direction. SI unit of velocity is meter/second.

Given that:

Mass of the ball: M = 6.35 kg.

Initial velocity of ball: U = 8.49 m/s.

Mass of the pin at rest: m = 1.59 kg.

Final velocity of pin: v = 20.1 m/s at a -77.0° angle.

Let the x and y component of the ball's final velocity are respectively V₁ m/s and  V₂ m/s.

Appling conservation of momentum along x axis:

MU + m.0 = MV₁ + mvcos(-77.0°)

⇒ V₁ = u - (m/M) v cos(-77.0°)

After putting the values we get:

V₁ = 7.35 m/s.

Appling conservation of momentum along y-axis:

M.0 + m.0 = MV₂ + mvsin(-77.0°)

⇒ V₂ = - (m/M) vsin(-77.0°)

After putting the values we get:

V₂ = 4.90 m/s.

Hence, the x and y component of the ball's final velocity are respectively 7.35 m/s and  4.90 m/s.

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Answer: They’re the most reactive metals

Explanation: They have larger ionic radii and low ionization energies

Indeed, you can run this experiment by adjusting the length (1) of the thread.

The pendulum's period is influenced by the length of the thread; the longer the string, the longer the pendulum's period. The frequency of the pendulum, or how quickly it swings back and forth, is also impacted by this.

The variables for plotting the graph would depend on the specific experiment and the data being collected. However, some possible variables that could be plotted on a graph include:

Length of the thread on the x-axis

Measured behavior or characteristic of the object or system on the y-axis

Constants or controlled variables, such as the mass or shape of the object being measured, on the graph as well.

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Answer:

tick = 30

Explanation:

The angular velocity in a simple pendulum is

          w² = g / L

the angular velocity is related to the period

          w = 2π/ T

let's substitute

          \(\frac{4 \pi }{T^2} = \frac{g}{L}\)

           T = \(2 \pi \sqrt{\frac{L}{g} }\)

indicate that the length is m = 1, if we assume that the acceleration of gravity is g = 9.8 m / s²

           T = 2π  \(\sqrt{\frac{1}{9.8} }\)

           T = 2.0 s

the clock has a tick each period and since the period is 2.0 s, to complete the minute (60 s) you need

           tick = 1 60 / T

           tick = 1 60/2

           tick = 30

The clock ticks 30 times in a minute.

Given that the clock ticks only when the pendulum makes one full swing and complete on cycle and returns to its original position. To calculate how many times the clock ticks we need to calculate the time period of the pendulum swing.

For a pendulum, the time period (T) is given by:

\(T=2\pi \sqrt\frac{L}{g}\)

where g is the acceleration due to gravity = 9.8\(m/s^{2}\),

and L is the length of the pendulum = 1m (given)

\(T=2*3.14 \sqrt\frac{1}{9.8}\)

\(T=2s\)

So, the pendulum takes 2 seconds to complete a cycle.

2s = 1 tick

60 second = 60/2 = 30 ticks

It will tick 30 times in a minute.

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The mass of the object is approximately 0.457 kg.

The mass of the object attached to the trolley can be calculated using the principle of conservation of momentum. Since the two trolleys couple together and move as a single system after the collision, the total momentum before and after the collision should be the same. Given the mass of one trolley is 0.80 kg and the initial speed is 1.1 m/s, the momentum before the collision is 0.80 kg * 1.1 m/s = 0.88 kg·m/s. After the collision, the total mass is the sum of the two trolleys, and the final speed is 0.70 m/s.

Using the momentum equation, the mass of the object can be calculated as follows:

Total momentum before collision = Total momentum after collision

0.88 kg·m/s = (0.80 kg + mass of the object) * 0.70 m/s

Solving for the mass of the object, we get:

0.88 kg·m/s = (0.80 kg + mass of the object) * 0.70 m/s

0.88 kg·m/s = 0.56 kg + 0.70 kg * mass of the object

0.88 kg·m/s - 0.56 kg = 0.70 kg * mass of the object

0.32 kg = 0.70 kg * mass of the object

Dividing both sides by 0.70 kg, we find:

mass of the object = 0.32 kg / 0.70 kg = 0.457 kg

The two trolleys collide and couple together, the total momentum before the collision is equal to the total momentum after the collision according to the principle of conservation of momentum.

The momentum of an object is defined as the product of its mass and velocity. In this case, the mass of one trolley is known (0.80 kg) and the initial speed is given (1.1 m/s), allowing us to calculate the momentum before the collision.

After the collision, the two trolleys move together at a new speed (0.70 m/s). By setting the initial momentum equal to the final momentum and solving for the unknown mass of the object, we can find its value.

In the calculation, we subtract the masses of the two trolleys from the total mass in order to isolate the mass of the object.

Dividing the difference in momentum by the product of the known mass and the new speed, we obtain the mass of the object. In this case, the mass of the object is approximately 0.457 kg.

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The average acceleration of the motorcycle over the given distance is approximately 9.39 m/s².

To calculate the average acceleration of the motorcycle, we can use the formula:

Average acceleration = (final velocity - initial velocity) / time

First, let's convert the final velocity from km/h to m/s since the distance is given in meters. We know that 1 km/h is equal to 0.2778 m/s.

Converting the final velocity:

Final velocity = 78 km/h * 0.2778 m/s = 21.67 m/s

Since the motorcycle starts from rest (initial velocity is zero), the formula becomes:

Average acceleration = (21.67 m/s - 0 m/s) / time

To find the time taken to reach this velocity, we need to use the formula for average speed:

Average speed = total distance/time

Rearranging the formula:

time = total distance / average speed

Plugging in the values:

time = 50 m / 21.67 m/s ≈ 2.31 seconds

Now we can calculate the average acceleration:

Average acceleration = (21.67 m/s - 0 m/s) / 2.31 s ≈ 9.39 m/s²

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Two forces f1=(8i+3j)N and f2=(4i+6j) are acting on 5kg object then  the magnitude of the resultant force is 15 N and the direction of the resultant force is approximately 36.87 degrees from the positive x-axis.

The acceleration of the object in the x-component (\(a_x\)) is 2.4  \(m/s^{2}\), and the acceleration in the y-component (\(a_y\)) is 1.8  \(m/s^{2}\).

The magnitude of the acceleration of the object is 3  \(m/s^{2}\).

To find the magnitude and direction of the resultant force, we need to add the two given forces together.

Given:

f1 = (8i + 3j) N

f2 = (4i + 6j) N

To find the resultant force (\(F_res\)), we simply add the corresponding components:

\(F_res\) = f1 + f2

= (8i + 3j) + (4i + 6j)

= (8 + 4)i + (3 + 6)j

= 12i + 9j

The magnitude of the resultant force (\(|F_res|\)) can be found using the Pythagorean theorem:

\(|F_res|\)= \(\sqrt{(12^2) + (9^2)}\)

= \(\sqrt{144 + 81}\)

= \(\sqrt{225}\)

= 15 N

So, the magnitude of the resultant force is 15 N.

To find the direction of the resultant force, we can use trigonometry. The direction can be represented by the angle θ between the positive x-axis and the resultant force vector. We can calculate θ using the inverse tangent function:

θ = arctan(9/12)

= arctan(3/4)

≈ 36.87 degrees

Therefore, the direction of the resultant force is approximately 36.87 degrees from the positive x-axis.

Now let's calculate the acceleration of the object in the x and y components. We know that force (F) is related to acceleration (a) through Newton's second law:

F = ma

For the x-component:

\(F_x\)= 12 N

m = 5 kg

Using  \(F_x\)= \(ma_x\), we can solve for \(a_x\):

12 N = 5 kg * \(a_x\)

\(a_x\)= 12 N / 5 kg

\(a_x\) = 2.4 \(m/s^{2}\)

For the y-component:

\(F_y\) = 9 N

m = 5 kg

Using \(F_y\) = \(ma_y\), we can solve for \(a_y\):

9 N = 5 kg * \(a_y\)

\(a_y\) = 9 N / 5 kg

\(a_y\)= 1.8  \(m/s^{2}\)

So, the acceleration of the object in the x-component (\(a_x\)) is 2.4  \(m/s^{2}\), and the acceleration in the y-component (\(a_y\)) is 1.8  \(m/s^{2}\).

To find the magnitude of the acceleration (|a|), we can use the Pythagorean theorem:

|a| = \(\sqrt{(a_x^2) + (a_y^2)}\)

= \(\sqrt{(2.4^2) + (1.8^2}\)

= \(\sqrt{5.76 + 3.24}\)

= \(\sqrt{9}\)

= 3  \(m/s^{2}\)

Therefore, the magnitude of the acceleration of the object is 3  \(m/s^{2}\)

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The unit vector normal to the plane is determined as (i  - 4j  -  13k) / √186.  

Option  B.

The magnitude of the cross product of a→ and b→, is determined as follows;

|a→ × b→| = (3i +4j -k) × (i- 3j + k )

= [i       j       k]

  [3     4      -1]

  [1     -3       1]

The cross product of the vectors is calculated as follows;

= i (4 - 3) - j(3 - - 1)  + k (-9 - 4)

= i  - 4j  -  13k

The magnitude of the vector is calculated as follows;

|n| = √ (1² + 4² + 13²)

|n| = √186

The unit vector normal to the plane is calculated as follows;

n = (i  - 4j  -  13k) / √186

Thus, the unit vector normal to the plane is determined as (i  - 4j  -  13k) / √186.   The answer is B.

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Variation of the relative intensity \(I(\lambda)\) with the wavelength \(\lambda\) in a line is given by \($$I(\lambda) \propto e^{-\left(\frac{m c^2\left(\lambda-\lambda_0\right)^2}{2 k T \lambda_0^2}\right)}=\exp \left(-\left(\frac{m c^2\left(\lambda-\lambda_0\right)^2}{2 k T \lambda_0^2}\right)\right)$$\)

As, for non-relativistic thermal velocities, the Doppler shift in frequency will be: \($$f=f_0\left(1+\frac{v}{c}\right)$$\)

where f is the observed frequency,

f_0 is the rest frequency,

v is the velocity of the emitter towards the observer

c is the speed of light.

As any volume part of the radiating body will have a distribution of speeds both toward and away from the observer, which will have the overall effect of widening the seen line.

If \(I(v)dv=P_v(v)dv\) is the fraction of particles with velocity component v to (v+dv) along a line of sight

Then the value of the distribution of frequencies will be

\($$I(f) \mathrm{d} f=P_f(f) \mathrm{d} f=P_v\left(v_f\right) \frac{\mathrm{d} v}{\mathrm{~d} f} \mathrm{~d} f$$\)

where \($v_f=c\left(\frac{f}{f_0}-1\right)$\) is the velocity towards the observer corresponding to the shift of the rest frequency f0 to f.

So, we can write it as:

\($$P_f(f) \mathrm{d} f=\frac{c}{f_0} P_v\left(c\left(\frac{f}{f_0}-1\right)\right) \mathrm{d} f$$\)

As in the case of the thermal Doppler broadening, the velocity distribution is given by the Maxwell distribution:

\($$P_v(v)=\sqrt{\frac{m}{2 \pi k T}} e^{-\frac{m v^2}{2 k T}} \mathrm{~d} v$$\)

So,

\($$P_f(f) \mathrm{d} f=\frac{c}{f_0} \sqrt{\frac{m}{2 \pi k T}} e^{-\left(\frac{m\left[c\left(\frac{f}{f_0}-1\right)\right]^2}{2 k T}\right)} \mathrm{d} f$$\)

This can be simplified as

\($$P_f(f) \mathrm{d} f=\sqrt{\frac{m c^2}{2 \pi k T f_0^2}} e^{-\left(\frac{m c^2\left(f-f_0\right)^2}{2 k T f_0^2}\right)} \mathrm{d} f$$\)

While in the non-relativistic limit,

\(-\frac{f-f_0}{f_0} \approx \frac{\lambda-\lambda_0}{\lambda_0^2}.\)

So, from above,

we will get:

\($$I(\lambda) \propto e^{-\left(\frac{m c^2\left(\lambda-\lambda_0\right)^2}{2 k T \lambda_0^2}\right)}=\exp \left(-\left(\frac{m c^2\left(\lambda-\lambda_0\right)^2}{2 k T \lambda_0^2}\right)\right)$$\)

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The correct questions may be like

Through a small window in a furnace, which contains a gas at a high temperature T, the spectral lines emitted by the gas molecules are observed. Because of molecular motions, each spectral line exhibits Doppler broadening. Show that the variation of the relative intensity  \(I(\lambda)\) with wavelength \(\lambda\) in a line is given by \($$I(\lambda) \propto \exp \left\{-\frac{m c^2\left(\lambda-\lambda_0\right)^2}{2 \lambda_0^2 k T}\right\}$$\)

Answer:

6.9m/s

Explanation:

Given parameters:

Acceleration of the object  = 4m/s²

Distance = from x; 2m to x; 8m

Unknown:

Average velocity  = ?

Solution:

From the given parameters, we use the right motion equation to solve the problem.

   v² = u² + 2aS

v is the final velocity

u is the initial velocity

a is the acceleration

 S is the distance covered

 Distance  = 8m - 2m  = 6m

 Initial velocity  = 0m/s

The final velocity gives us the average velocity in this problem;

       v² = 0² + (2 x 4 x 6) = 48

      v = √48  = 6.9m/s

Option C. The heat absorbed or lost by a substance while it's changing state  correctly describes latent heat

Latent heat is the heat absorbed or lost by a substance while it is changing state.

The latent heat is a type of heat that is transferred during phase change, i.e., while a substance undergoes a change of state.

For example, when ice melts into liquid water, or when liquid water evaporates into water vapor, heat is absorbed from the surroundings.

Latent heat is not associated with a temperature change; rather, it's associated with a change of state.

For instance, the temperature of water remains at 100°C while boiling.

When water is boiling, the latent heat of vaporization is absorbed and utilized to break the hydrogen bonds holding water molecules together to change water from the liquid phase to the gaseous phase.

When the water is boiling, adding more heat won't increase the water's temperature, instead, the extra heat will be absorbed to change the phase of water molecules.

Therefore, the correct answer to the given question is option C: The heat absorbed or lost by a substance while it is changing state.

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To find the mass in kilograms, you need to know the object's weight in newtons and the acceleration due to gravity. The formula for finding mass is mass = weight / acceleration due to gravity. So if you have an object with a weight of 100 N and the acceleration due to gravity is 9.8 m/s^2, the mass would be 10.204 kg.

The mass of the block is 0.025 kg or 25 g, when the spring has k = 28 N/m, and compresses 0.11 m before bringing the block to rest.

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Answer:

Step 1: List the mass and velocity of the object. Step 2: Convert any values into SI units (kg, m, s). Step 3: Multiply the mass and velocity of the object together to get the momentum of the object.

Answer:

un café y a seguir pensando porque no se la respuesta xd

Because the weight of the helium gas is less than the weight of the air it has displaced from the volume of the balloon, the helium-filled balloon rises in the atmosphere.

Chemical element helium has the atomic number 2 and the symbol He. It is the first member of the noble gas group in the periodic table and is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas. Its melting point at ordinary pressure is zero, and its boiling point is the lowest of all the elements.

Helium is most commonly used to fill balloons for celebrations and parades since it is a secure, non-flammable gas. Helium, however, is an essential component in numerous industries, including as national security, high-tech manufacturing, medical technology, and scientific research.

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Answer:

natural convection

............

Answer:

See below

Explanation:

Frequency = 1 / period

                  = 1 / 460 X 10^-6

            2173.91 Hz    = ~ 2.17 Khz

2.17391 kHz / 4.25 = .511 kHz   <====new frequency

     period = 1/frequency

                     = 1955 microseconds =1.96 ms

Answer:

Explanation:

You want to know the frequency in kHz of a sine wave with a period of 460 μs, and its period if the frequency is reduced by a factor of 4.25.

When you are interested in the frequency in kHz, it is convenient to use milliseconds (ms) to express the period. Milliseconds and kilohertz are inverse units.

The period of the given sine wave can be expressed in milliseconds as ...

  460 μs = 460×(0.001 ms) = 0.460 ms

The frequency of the wave is 1/(0.460 ms) ≈ 2.17 kHz.

Because period and frequency are reciprocals of each other, reducing the frequency by a factor of 4.25 increases the period by the same factor.

The new period is 4.25×0.460 ms = 1.96 ms.

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The motorcycle had covered a distance of 16 meters in half the total time.

a) To calculate the acceleration, we can use the formula:

a = (v - u) / t

where a is the acceleration, v is the final velocity, u is the initial velocity (which is 0 since the motorcycle starts from rest), and t is the time.

Given:

u = 0 m/s (initial velocity)

v = ? (final velocity)

t = 4 s (time)

s = 64 m (distance)

Using the equation of motion:

s = ut + 1/2at^2

We can rearrange the equation to solve for acceleration:

a = 2s / t^2

a = 2(64) / (4)^2

a = 128 / 16

a = 8 m/s^2

Therefore, the acceleration of the motorcycle is 8 m/s^2.

b) To find the final velocity, we can use the formula:

v = u + at

v = 0 + (8)(4)

v = 32 m/s

Therefore, the final velocity of the motorcycle is 32 m/s.

c) To determine the time at which the motorcycle had covered half the total distance, we divide the total distance by 2 and use the formula:

s = ut + 1/2at^2

32 = 0 + 1/2(8)t^2

16 = 4t^2

t^2 = 4

t = 2 s

Therefore, the motorcycle had covered half the total distance at 2 seconds.

d) To calculate the distance covered in half the total time, we use the formula:

s = ut + 1/2at^2

s = 0 + 1/2(8)(2)^2

s = 0 + 1/2(8)(4)

s = 0 + 16

s = 16 m

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A cell of inter resistance of 0.5 ohm is connected to coil of resistance 4 ohm and 8 ohm joined in parallel.If there is current of 2A in 8 ohm, the electromotive force (emf) of the cell is approximately 14.5 volts.

To find the emf of the cell, we can apply Ohm's Law and Kirchhoff's laws to analyze the circuit.

Given:

Resistance of the coil, R1 = 4 ohm

Resistance of the other resistor, R2 = 8 ohm

Current passing through the 8-ohm resistor, I = 2A

First, let's analyze the parallel combination of the 4-ohm and 8-ohm resistors.

The total resistance of two resistors in parallel can be calculated using the formula:

1/Rp = 1/R1 + 1/R2

Substituting the given values, we have:

1/Rp = 1/4 + 1/8

1/Rp = 2/8 + 1/8

1/Rp = 3/8

Rp = 8/3 ohm

Now, let's consider the total resistance in the circuit, which includes the internal resistance of the cell (0.5 ohm) and the parallel combination of the resistors (8/3 ohm).

R_total = R_internal + Rp

R_total = 0.5 + 8/3

R_total = 1.833 ohm

Now, we can find the emf of the cell using Ohm's Law:

emf = I * R_total

emf = 2 * 1.833

emf ≈ 3.667 volts

Therefore, the emf of the cell is approximately 3.667 volts.

However, it is worth noting that the given current of 2A passing through the 8-ohm resistor does not affect the emf calculation since the emf of the cell is independent of the current in the circuit.

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Answer:

Broadcasting is the method, not sure about the stage it is done in

Explanation:

Answer:

Broadcasting is the first (blank) second (blank) is Cultivation.

Explanation:

I took the test & got this answer correct.

If we know the mass of cart and its acceleration, we can calculate the magnitude of force that the cart exerts on the horse (which is equal in magnitude to the force that horse exerts on cart).

An external agent that is capable of changing body's state of rest or motion is known as the force.

According to third law of motion of Newton, for every action, there is equal and opposite reaction. In this case,  horse is exerting a force on the cart to accelerate it, and by Newton's third law, the cart exerts an equal and opposite force on horse.

Therefore, the magnitude of force that cart exerts on the horse is equal to the magnitude of the force that horse exerts on the cart. This force is equal to the mass of cart multiplied by its acceleration (assuming negligible friction and air resistance): F = m*a

F is the force exerted by the horse on cart (and vice versa), m is mass of  cart, and a is acceleration.

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The effort distance​ will be 160 cm.Applying the moment at the center as follows will provide the effort distance:

Mechanical advantage is a measure of the ratio of output force to input force in a system, it is used to obtain the efficiency of forces in levers and pulleys.

Given data;

Effort,\(\rm F_e=00 N\)

Load,\(\rm P= 400 \ N\)

Distance from the fulcrum,\(\rm d=40 \ cm\)

The effort distance​ is found by applying the moment at the center as;

\(\rm F_e \times d= P \times d' \\\\ 200 \ N \times d'=800 \ N \times 40 \ cm \\\\ d'=160 \ cm\)

Hence, the effort distance​ will be 160 cm

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Rotational KE is the energy of a rotating object. For a CD with a mass of 0.0140kg, a radius of 0.0600m, and an angular velocity of 31.4 rad/s, the rotational KE is 0.0186 J.

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Answer:

17

Explanation:

2 x 10 = 20

20-3 = 17

The net force on q₂ will be  1.07 x 10⁻² N, pointing to the left.

To find the net force on particle q₂, we need to calculate the force due to q₁  and q₃ individually and then add them up vectorially. We can use Coulomb's law to calculate the force between two point charges:

F = k × (q₁ × q₂) / r²

where F is the magnitude of the force, k is Coulomb's constant (k = 8.99 x 10⁹ Nm²/C²), q1 and q2 are the charges of the two particles, and r is the distance between them.

The force due to q₁ on q₂ can be calculated as:

F₁ = k × (q₁ × q₂) / r₁²

where r1 is the distance between q₁ and q₂ (r₁ = 0.180 m).

Similarly, the force due to q₃ on q₂ can be calculated as:

F₂ = k × (q₃ × q₂) / r₃²

where r₃ is the distance between q₂ and q₃ (r₃= 0.230 m).

The direction of each force can be determined by the direction of the electric field due to each charge. Since q₁ and q₃ have opposite signs, their electric fields point in opposite directions. Therefore, the force due to q₁ points to the left and the force due to q₃ points to the right.

To find the net force, we need to add up the forces vectorially. Since the forces due to q₁ and q₃ are in opposite directions, we can subtract the magnitude of the force due to q₃ from the magnitude of the force due to q₁ to get the net force on q₂:

Fnet = F₁ - F₃

Substituting the values we get:

Fnet = k × (q₁ × q₂) / r₁² - k × (q₃ × q₂) / r₃²

Plugging in the values we get:

Fnet = (8.99 x 10⁹ Nm²/C²) × [(9.33 x 10⁻⁶ C) × (4.22 x 10⁻⁶ C) / (0.180 m)² - (-8.42 x 10⁻⁶ C) × (4.22 x 10⁻⁶ C) / (0.230 m)²]

Fnet = 1.07 x 10⁻² N

Therefore, the net force on q₂ is 1.07 x 10⁻² N, pointing to the left.

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The equivalent resistance, Rₜ can be obtained as illustrated below:

Rₜ = (R₁ × R₂) / (R₁ + R₂)

= (10 × 20) / (10 + 20)

= 200 / 30

= 6.67 Ω

The total current, Iₜ can be obtained as follow:

Current = Voltage / resistance

= 90 / 6.67

= 13.5 A

The voltage, V₁ and V₂ can be obtained as follow:

Voltage in parallel connection is the same through out the circuit.

Thus,

Vₜ = V₁ = V₂ (parallel connection)

90 = V₁ = V₂

Therefore,

V₁ = V₂ = 90 V

We can obtain current, I₁ as follow:

I₁ = V₁ / R₁

= 90 / 10

= 9 A

We can obtain current, I₂ as follow:

I₂ = V₂ / R₂

= 90 / 20

= 4.5 A

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The extension in the string when the particle hangs in equilibrium is e = mg/k. The constant n in the equation of motion is n = (2π/T)²m. The time it takes for the particle to enter simple harmonic motion is t = √(2§/g).

To find the extension in the string when the particle hangs in equilibrium, we need to consider the forces acting on the particle. The weight of the particle is mg, acting vertically downward. The tension in the string exerts an equal and opposite force, mg, to balance the weight. Since the particle is in equilibrium, the total force acting on it is zero.When the particle is dropped from a point below O, it will stretch the string due to its weight. Let's denote the extension of the string as e. The tension in the string will now be greater than mg because it needs to support the weight of the particle and provide the necessary centripetal force for the simple harmonic motion.The extension in the string is given by Hooke's law: F = kx, where F is the force, k is the modulus of the string, and x is the extension. In this case, the force is the tension in the string, and the extension is e. Thus, we have mg = ke. Rearranging the equation, we get e = mg/k.

To find the constant n in the equation of motion, › = -n?, we can use the fact that the period of simple harmonic motion is given by T = 2π√(m/k), where m is the mass of the particle and k is the spring constant. Comparing this with the equation of motion, we see that ω² = n/m. Since ω = 2π/T, we can substitute ω in terms of T and solve for n: n = (2π/T)²m.

To determine when the particle enters simple harmonic motion, we need to find the time it takes for the string to reach its equilibrium position. Since the particle is dropped from a point § meters below O, it will accelerate towards the equilibrium position due to the tension in the string. The time taken for this free-fall motion can be calculated using the equation of motion for constant acceleration: § = ½gt². Solving for t, we get t = √(2§/g).The amplitude of the simple harmonic motion is equal to the maximum displacement from the equilibrium position. In this case, the maximum displacement is equal to the initial dropped distance, § meters.To determine the time it takes for the particle to come to instantaneous rest for the first time, we need to find the time period of the simple harmonic motion. The time period, T, is given by T = 2π√(m/k). To find the time it takes to come to rest, we need to consider half of the time period, T/2.The amplitude of the simple harmonic motion is equal to the initial dropped distance, § meters. The time it takes for the particle to come to instantaneous rest for the first time is T/2, where T is the time period of the simple harmonic motion.

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