If a material has an index of refraction of 1.61, Determine the speed of light through this medium

Answer:

The acceleration of each of the two blocks and the tension in the rope are 5.92 m/s and 147.44 N.

Explanation:

Given that,

Mass = 12.0 kg

Angle = 31.0°

Friction coefficient = 0.158

Mass of second block = 38.0 kg

Using formula of frictional force

[tex]f_{\mu} = \mu N[/tex]....(I)

Where, N = normal force

[tex]N = mg\cos\theta[/tex]

Put the value of N into the formula

[tex]N =12\times9.8\times\cos 31^{\circ}[/tex]

[tex]N=100.80\ N[/tex]

Put the value of N in equation (I)

[tex]f_{mu}=0.158\times100.80[/tex]

[tex]f_{mu}=15.9264\ N[/tex]

Now, Weight of second block

[tex]W = mg[/tex]

[tex]W=38.0\times9.8[/tex]

[tex]W=372.4\ N[/tex]

The horizontal force is

[tex]F = mg\sintheta[/tex]

[tex]F=12\times9.8\times\sin 31^{\circ}[/tex]

[tex]F=60.5684\ N[/tex]....(II)

(I). We need to calculate the acceleration

[tex]a=m_{2}g-\dfrac{f_{\mu}+mg\sin\theta}{m_{1}+m_{2}}[/tex]

[tex]a=\dfrac{372.4-(15.9264+60.5684)}{12+38}[/tex]

[tex]a=5.92\ m/s^2[/tex]

(II). We need to calculate the tension in the rope

[tex]m_{2}g-T=m_{2}a[/tex]

[tex]-T=38\times5.92-38\times9.8[/tex]

[tex]T=147.44\ N[/tex]

Hence, The acceleration of each of the two blocks and the tension in the rope are 5.92 m/s and 147.44 N.

Answer:

2.08 x 10⁻⁶ T

Explanation:

[tex]v[/tex] = velocity of electron = 5.0 x 10⁷ m/s

q = charge on electron

B = magnetic field = ?

E = electric field = 104 V/m

Magnetic force on the electron is given as

[tex]F_{B} = qvB[/tex]

Electric force on the electron is given as

[tex]F_{E} = qE[/tex]

For the electron to pass without being deflected, we must have

[tex]F_{B} = F_{E}[/tex]

[tex]qvB = qE[/tex]

[tex]vB = E[/tex]

(5.0 x 10⁷) B = 104

B = 2.08 x 10⁻⁶ T

Answer:

3272.4 Hz

Explanation:

L = 31 mH

XL = 637 ohm

XL = 2 π f L

f = XL / (2 π L)

f = 637 / ( 2 x 3.14 x 31 x 10^-3)

f = 3272.4 Hz

Answer:

Total energy, [tex]TE=3.006\times 10^{-10}\ J[/tex]

Explanation:

It is given that, a proton in a certain particle accelerator has a kinetic energy that is equal to its rest energy.  Let KE is the kinetic energy of the proton and E₀ is its rest energy. So,

[tex]KE=E_o[/tex]              

The total energy of the proton is equal to the sun of kinetic energy and the rest mass energy.

[tex]TE=KE+E_o[/tex]

[tex]TE=2E_o[/tex]

[tex]TE=2m_{proton}c^2[/tex]

[tex]TE=2\times 1.67\times 10^{-27}\ kg\times (3\times 10^8\ m/s)^2[/tex]

[tex]TE=3.006\times 10^{-10}\ J[/tex]

So, the total energy of the proton as measured by a physicist working with the accelerator is [tex]3.006\times 10^{-10}\ J[/tex]

Answer:

(a) 7.92 A

(b) 0.133 Nm

Explanation:

N = 216

r = 2.32 cm = 0.0232 m

(a) M = 2.89 Am^2

M = N i A

Where, A be the area of the coil and i be the current in the coil

2.89 = 216 x i x 3.14 x 0.0232 x 0.0232

i = 7.92 A

(b) B = 46 mT = 0.046 T

Torque, τ = M B Sin 90

τ = 2.89 x 0.046 x 1 = 0.133 Nm

The current required to produce a magnetic dipole moment of 2.89 A·m² in a coil of 216 turns and radius 2.32 cm is approximately 7.92 A. The maximum torque this coil can experience in a 46.0 mT magnetic field is about 0.133 N·m.

To solve these problems, we'll use the relationship between the magnetic dipole moment, current, and torque in a magnetic field.

Part (a): Calculate the Current

The magnetic dipole moment (μ) of a coil is given by:

μ = NIA

where N is the number of turns, I is the current, and A is the area of the coil.

Given:

N = 216 turns

μ = 2.89 A·m²

Radius (r) = 2.32 cm = 0.0232 m

The area (A) of the coil is:

A = πr² = π(0.0232 m)² ≈ 1.69 × 10⁻³ m²

To find the current (I), rearrange the formula:

I = μ / (N × A)

Substitute the values:

I = 2.89 A·m² / (216 × 1.69 × 10⁻³ m²) ≈ 7.92 A

Part (b): Calculate the Maximum Torque

The maximum torque (τ) experienced by the coil in a magnetic field (B) is given by:

τ = μB

Given the magnetic field (B) is 46.0 mT = 0.046 T, substitute the values:

τ = 2.89 A·m² × 0.046 T ≈ 0.133 N·m

Thus, the current required to achieve the given magnetic dipole moment is approximately 7.92 A, and the maximum torque experienced by the coil in a 46.0 mT magnetic field is approximately 0.133 N·m.

Answer:

[tex]B_2 = 1.71 mT[/tex]

Explanation:

As we know that the magnetic field near the center of solenoid is given as

[tex]B = \frac{\mu_0 N i}{L}[/tex]

now we know that initially the length of the solenoid is L = 18 cm and N number of turns are wounded on it

So the magnetic field at the center of the solenoid is 2 mT

now we pulled the coils apart and the length of solenoid is increased as L = 21 cm

so we have

[tex]\frac{B_1}{B_2} = \frac{L_2}{L_1}[/tex]

now plug in all values in it

[tex]\frac{2.0 mT}{B_2} = \frac{21}{18}[/tex]

[tex]B_2 = 1.71 mT[/tex]

The magnetic field strength inside a solenoid is dependent on the density of the coils per unit length. If a solenoid is extended, spreading the same number of coils over a greater distance, the magnetic field strength at its center decreases proportionally.

The magnetic field strength (B) inside a solenoid is proportional to the number of coils per unit length (n) and the current flowing through the coils (I). We can express this mathematically as B = μonI. The key concept here is that the field strength, B, depends on the density of the coils (n); in other words, how tightly or loosely coiled the wires are.

When you initially have a solenoid 18 cm long, and you extend it to 21 cm without changing the current or the size of the coils, the coil density per unit length decreases. Therefore, the magnetic field strength should decrease proportionally. In effect, you are spreading the same number of coils (field lines) over a greater distance, which dilutes the strength of the magnetic field at the solenoid's center. The exact strength would require knowing more specifics about the solenoid, but this discussion gives the general principle.

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Answer:

The width of the slit is 0.167 mm

Explanation:

Wavelength of light, [tex]\lambda=608\ nm=608\times 10^{-9}\ m[/tex]

Distance from screen to slit, D = 88.5 cm = 0.885 m

The distance on the screen between the fifth order minimum and the central maximum is 1.61 cm, y = 1.61 cm = 0.0161 m

We need to find the width of the slit. The formula for the distance on the screen between the fifth order minimum and the central maximum is :

[tex]y=\dfrac{mD\lambda}{a}[/tex]

where

a = width of the slit

[tex]a=\dfrac{mD\lambda}{y}[/tex]

[tex]a=\dfrac{5\times 0.885\ m\times 608\times 10^{-9}\ m}{0.0161\ m}[/tex]

a = 0.000167 m

[tex]a=1.67\times 10^{-4}\ m[/tex]

a = 0.167 mm

So, the width of the slit is 0.167 mm. Hence, this is the required solution.

Answer:

Work, W = 846.72 Joules

Explanation:

Given that,

Mass of the watermelon, m = 4.8 kg

It is dropped from rest from the roof of 18 m building. We need to find the work done by the gravity on the watermelon from the roof to the ground. It is same as gravitational potential energy i.e.

W = mgh

[tex]W=4.8\ kg\times 9.8\ m/s^2\times 18\ m[/tex]

W = 846.72 Joules

So, the work done by the gravity on the watermelon is 846.72 Joules. Hence, this is the required solution.

Answer:

W = 86 J

Explanation:

Work done by friction force is given by the formula

[tex]W = F_f . d[/tex]

here since the wagon is pulled by horizontal force such that speed remains constant

so here we will have

[tex]F_f = F_{ext} = 22 N[/tex]

Now we will have

[tex]W = 22\times 3.9 [/tex]

[tex]W = 86 J[/tex]

so work done against friction force will be equal to 86 J

The correct answer is option c 86.

Calculating Work Done by the Child:

In this  problem, we need to determine the amount of work done by a child who is pulling a 12 kg wagon over a distance of 3.9 meters with a force of 22 Newtons, assuming there is no friction.

The formula to calculate work (W) is:

W = F * d

where W is work, F is the force applied, and d is the distance covered.

Let's plug in the values provided:

F = 22 N
d = 3.9 m

Thus, the work done is:

W = 22 N * 3.9 m

W = 85.8 J

Therefore, the correct answer is approximately 86 J.

Answer:

5.3 x 10⁻⁹ C

Explanation:

r = radius of cylindrical shell = 10⁻⁵ m

L = length = 0.32 m

A = area

Area is given as

A = 2πrL

A = 2 (3.14) (10⁻⁵) (0.32)

A = 20.096 x 10⁻⁶ m²

d = separation = 10⁻⁸ m

[tex] k_{appa} [/tex] = dielectric constant = 4

Capacitance is given as

[tex]Q=\frac{k_{appa}\epsilon _{o}A}{d}[/tex]                               eq-1

V = Potential difference across the membrane = 74 mV = 0.074 Volts

Q = magnitude of charge on each side

Magnitude of charge on each side is given as

Q = CV

using eq-1

[tex]Q=\frac{k_{appa} \epsilon _{o}AV}{d}[/tex]

Inserting the values

[tex]Q=\frac{4 (8.85\times 10^{-12})(20.096\times 10^{-6})(0.074)}{10^{-8}}[/tex]

Q = 5.3 x 10⁻⁹ C

The membrane is parallel and act as a parallel plate capicitor.The magnitude of the charge on each side of the membrane will be 5.3×10⁻⁹C.

It is an type capacitor is an in which two metal plates arranged in such away so that they are connected in parallel andhaving some distance between them.

A dielectric medium is must in between these plates help to stop the flow of electric current through it due to its non-conductive nature .

The given data in the problem is;

r is the radius of cell of thin cyliner=10⁻⁵m

L is the length =0.32

d is the thickness =10⁻⁸m

K is the dielectric constant=4

v is the potential difference across the membrane=74 mv=0.074 v

q is the megnitude of charge=?

A is the area of capicitor=2πrl

[tex]\rm A = 2\pi rl\\\\\rm A= 2\times3.14\times10^{-5}\times0.32\\\\\rm A= 20.096\times10^{-6}[/tex]

The given formula for the parallel plate capicitor as ,

[tex]\rm Q=\frac{K\varepsilon _0AEr}{d} \\\\\rm Q=\frac{4\times(8.85\times10^{-12})(20.096\times10^{-6}\times0.074}{10^{-8}}\\\\\rm Q=5.3\TIMES10^{-9}\;C[/tex]

Hence the charge on each side of the membrane will be 5.3×10⁻⁹C.

To learn more about the parallel plate capicitor refer to the link;

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Explanation:

It is given that,

Electric field, E = 554 N/C

Time, [tex]t=52\ ns=52\times 10^{-9}\ s[/tex]

Electric force, F = qE

For both electron and proton, [tex]F=1.6\times 10^{-19}\ C\times 554\ N/C[/tex]

[tex]F=8.86\times 10^{-17}\ N[/tex]

For electron, [tex]F=m_ea_e[/tex]

[tex]a_e=\dfrac{F}{m_e}[/tex]

[tex]a_e=\dfrac{8.86\times 10^{-17}\ N}{9.1\times 10^{-31}\ kg}[/tex]

[tex]a_e=9.73\times 10^{13}\ m/s^2[/tex]

Using first equation of motion as :

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

u = 0

[tex]v=9.73\times 10^{13}\ m/s^2\times 52\times 10^{-9}\ s[/tex]

v = 5059600 m/s

or

v = 5.05 × 10⁶ m/s

For proton :

[tex]F=m_pa_p[/tex]

[tex]a_p=\dfrac{F}{m_e}[/tex]

[tex]a_p=\dfrac{8.86\times 10^{-17}\ N}{1.67\times 10^{-27}\ kg}[/tex]

[tex]a_p=5.3\times 10^{10}\ m/s^2[/tex]

Using first equation of motion as :

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

u = 0

[tex]v=5.3\times 10^{10}\ m/s^2\times 52\times 10^{-9}\ s[/tex]

v = 2756 m/s

Hence, this is the required solution.

Answer:

F = - 59.375 N

Explanation:

GIVEN DATA:

Initial velocity = 11 m/s

final velocity = 1.5 m/s

let force be F

work done =  mass* F = 4*F

we know that

Change in kinetic energy = work done

kinetic energy = [tex]= \frac{1}{2}*m*(v_{2}^{2}-v_{1}^{2})[/tex]

kinetic energy = [tex]= \frac{1}{2}*4*(1.5^{2}-11^{2})[/tex] = -237.5 kg m/s2

-237.5 = 4*F

F = - 59.375 N

The magnitude of the non-conservative force acting on the box is 236 J and it acts horizontally.

The magnitude and direction of the non-conservative force acting on the box can be determined using the work-energy theorem. The work done by the non-conservative force is equal to the change in kinetic energy of the box. From the given information, the initial kinetic energy of the box is 0.5 * 4.0 kg * (11 m/s)^2 = 242 J and the final kinetic energy is 0.5 * 4.0 kg * (1.5 m/s)^2 = 6 J. Therefore, the work done by the non-conservative force is 242 J - 6 J = 236 J. Since the box moves horizontally, the non-conservative force acts horizontally as well.

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To find the force exerted by the ground on each wheel of the automobile, we can analyze the forces acting on the car and use the principle of equilibrium. By considering the weight of the car and the distribution of weight between the front and rear axles, we can determine the force exerted by the ground on each wheel.

The force exerted by the ground on each wheel of the automobile can be determined by considering the forces acting on the car. Since the car is in equilibrium, the sum of the vertical forces must be zero. The weight of the car is distributed between the front and rear axles according to their distances from the center of mass. Using this information, we can calculate the force exerted by the ground on each wheel.

First, we find the weight of the car by multiplying its mass by the acceleration due to gravity: W = (mass of the car) x (acceleration due to gravity). In this case, the acceleration due to gravity is 9.8 m/s².

Next, we find the force exerted by the ground on the rear wheels. Since the car is in equilibrium, the sum of the torques about any point must be zero. Taking the point where the front wheels contact the ground as the pivot point, we can set up an equation using the weight of the car, the distances between the front and rear axles and the center of mass, and the force exerted by the ground on the rear wheels. Solving this equation allows us to find the force exerted by the ground on the rear wheels, and since the front and rear wheels share the weight of the car equally, the force exerted by the ground on each wheel is half of this value.

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Answer:

5.4 x 10⁶ Pa

Explanation:

h = depth to which penguins dive under seawater = 530 m

P₀ = Atmospheric pressure = 1.013 x 10⁵ pa

ρ = density of seawater = 1025 kg/m³

P = absolute pressure experienced by penguin at that depth

Absolute pressure is given as

P = P₀ + ρgh

Inserting the values

P = 1.013 x 10⁵ + (1025) (9.8) (530)

P = 5.4 x 10⁶ Pa

Answer:

Approximately 18 volts when the magnetic field strength increases from [tex]\rm 20\; mT[/tex] to [tex]\rm 80\;mT[/tex] at a constant rate.

Explanation:

By the Faraday's Law of Induction, the EMF [tex]\epsilon[/tex] that a changing magnetic flux induces in a coil is:

[tex]\displaystyle \epsilon = N \cdot \frac{d\phi}{dt}[/tex],

where

However, for a coil the magnetic flux [tex]\phi[/tex] is equal to

[tex]\phi = B \cdot A\cdot \cos{\theta}[/tex],

where

For this coil, the magnetic field is perpendicular to coil, so [tex]\theta = 0[/tex] and [tex]A\cdot \cos{\theta} = A[/tex]. The area of this circular coil is equal to [tex]\pi\cdot r^{2} = \pi\times 8.0\times 10^{-2}\approx \rm 0.0201062\; m^{2}[/tex].

[tex]A\cdot \cos{\theta} = A[/tex] doesn't change, so the rate of change in the magnetic flux [tex]\phi[/tex] through the coil depends only on the rate of change in the magnetic field strength [tex]B[/tex]. The size of the magnetic field at the instant that [tex]B = \rm 50\; mT[/tex] will not matter as long as the rate of change in [tex]B[/tex] is constant.

[tex]\displaystyle \begin{aligned} \frac{d\phi}{dt} &= \frac{\Delta B}{\Delta t}\times A \\&= \rm \frac{80\times 10^{-3}\; T- 20\times 10^{-3}\; T}{20\times 10^{-3}\; s}\times 0.0201062\;m^{2}\\&= \rm 0.0603186\; T\cdot m^{2}\cdot s^{-1}\end{aligned}[/tex].

As a result,

[tex]\displaystyle \epsilon = N \cdot \frac{d\phi}{dt} = \rm 300 \times 0.0603186\; T\cdot m^{2}\cdot s^{-1} \approx 18\; V[/tex].

Using Faraday's law, the magnitude of the induced emf in the coil at the instant the magnetic field is 50 mT is found to be 18 volts. The negative sign indicates the emf opposes the change in flux according to Lenz's law.

The question is about the calculation of the magnitude of induced emf in the coil at the instant the magnetic field has a magnitude of 50 mT. To approach this, we can use Faraday's law of electromagnetic induction. It states that the induced emf is equal to the rate of change of the magnetic flux.  

First, let us recall the expression of the magnetic flux (Φ): Φ = B * A, where B is the magnetic field's magnitude and A is the area through which it passes. For a circular coil, A = π * (radius)^2. Applying the given radius 0.08 m, we calculate A = 0.02 m².  

Given that the magnetic field changes linearly with time from 20 mT to 80 mT in 20 ms, we can determine the rate of change of the magnetic field which is (80 mT - 20 mT) / 20 ms = 3 T/s. Therefore, the rate of change of Flux is dΦ/dt = (B * A)/dt = 3 T/s * 0.02 m² = 0.06 Wb/s.

Finally, as per Faraday's law, the induced emf = -NdΦ/dt, where N is the number of turns in the coil. Therefore, for N = 300 turns, emf = -300 * 0.06 Wb/s = -18 V. The negative sign indicates the emf would oppose the change in flux according to Lenz's law.

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Answer:

the girl moves towards right with a velocity of 4.1m/s

Explanation:

Since the system is isolated the momentum of the system is conserved

Initial momentum = Final Momentum

Since initially the system is at rest thus [tex]\overrightarrow{p_{i}}=0[/tex]

Now the final momentum of boy = [tex]m_{boy}×velocity[/tex]

[tex]\overrightarrow{p_{boy}}=66.0\times -2.80m/s\\\\\overrightarrow{p_{boy}}=-184.8kgm/s[/tex]

Now for girl let the velocity = u hence her moumentum is 45[tex]\times u[/tex]

Thus equating final momentum to zero we have

[tex]-184.8kgm/s[/tex]+[tex]45\times u = 0[/tex]

[tex]u=\frac{184.8}{45}m/s[/tex]

hence [tex]u=4.1m/s[/tex]

Thus the girl moves towards right with a velocity of 4.1m/s

When the girl pushes her brother on roller blades, causing him to move backwards, the principle of conservation of momentum dictates that the girl will move in the opposite direction with a velocity of 4.16 m/s towards the east.

The scenario presented involves the conservation of momentum, which is a fundamental concept in physics. When the girl pushes her brother on roller blades, and he moves backward at a velocity of 2.80 m/s, by the principle of conservation of momentum, the girl will move in the opposite direction.

Since no external forces are acting on the system (assuming friction is ignored), the total momentum before and after the push remains constant. The combined momentum of the boy and girl before the push is zero because they are both initially at rest. After the push, the momentum of the boy is his mass multiplied by his velocity (66.0 kg × 2.80 m/s).

To find the subsequent motion of the girl, let's calculate:

Therefore, the girl will move towards the east at a velocity of 4.16 m/s as a result of the push.

Answer:

1196.62 sec

Explanation:

V = electric potential difference at which teapot operates = 120 volts

i = current = 2.00 A

t = time of operation

m = mass of water = 0.891 kg

T₀ = initial temperature = 23.0 °C

T = final temperature = 100 °C

c = specific heat of water = 4186 J/(Kg °C)

Using conservation of energy

V i t = m c (T - T₀)

(120) (2.00) t = (0.891) (4186) (100 - 23.0)

t = 1196.62 sec

Solution:

To calculate the average distance between the given parabola and the x-axis

y = 77x(1616 - x)

x ∈ [0, 16]

avg distance = [tex]\int_{0}^{16}\frac{77x(1616 - x)dx}{\int_{0}^{16} x dx}[/tex]

                     = [tex]2\int_{0}^{16}(\frac{(124432x - 77x^{2})dx}{[x^{2}]_{0}^{16}}[/tex]

                      =[tex] 2\int_{0}^{16}\frac{\frac{124432x^{2}}{2}- \frac{77x^{3}}{3}}{[x^{2}]_{0}^{16}}[/tex]

                      = [tex]16^{2}[\frac{62216 - 25.67\times 16}{16^{2}}][/tex]

avg distance = 61805 unit

Answer:

Electrical charge, q = 75 C

Explanation:

It is given that,

Voltage, V = 4 V

Energy stored, E = 300 J

Energy stored in the battery is given by :

[tex]E=q\times V[/tex]

q is the electrical charge

[tex]q=\dfrac{E}{V}[/tex]

[tex]q=\dfrac{300\ J}{4\ V}[/tex]

q = 75 C

So, the electrical charge of 75 C must flow between the battery's terminals to completely drain the battery if it is fully charged. Hence, this is the required solution.

Answer:

150 fils

Explanation:

Power = 125 Watt

Time 4 hours a day

Energy per day = Power x time per day = 125 x 4 = 500 watt hour

Energy for 30 days = 500 x 30 watt hour = 15000 watt hour = 15 KWH

Cost of 1 KWH = 10 fils

Cost of 15 KWH = 15 x 10 = 150 fils

Answer:

The length of the rectangular key is 0.4244 m

Explanation:

Given that,

Power = 7.46 kW

Speed = 1200 rpm

Shearing stress of shaft = 30 MPa

Mini shearing stress of key = 240 MPa

We need to calculate the torque

Using formula of power

[tex]P=\dfrac{2\pi NT}{60}[/tex]

Where, P = power

N = number of turns

Put the value into the formula

[tex]7.46\times10^{3}=\dfrac{2\pi\times1200\times T}{60}[/tex]

[tex]T=\dfrac{7.46\times10^{3}\times60}{2\pi\times1200}[/tex]

[tex]T=59.36\ N-m[/tex]

We need to calculate the distance

[tex]\tau_{max}=\dfrac{16T}{\pi d^3}[/tex]

[tex]d^3=\dfrac{16\times59.36}{\pi\times30}[/tex]

[tex]d=(10.077)^{\dfrac{1}{3}}[/tex]

[tex]d=2.159\ m[/tex]

Width of key is one fourth of the shaft diameter

[tex]W=\dfrac{1}{4}\times2.159[/tex]

[tex]W=0.53975\ m[/tex]

The shear stress induced in key

[tex]\tau_{max}=\dfrac{F}{Wl}[/tex]

[tex]\tau_{max}=\dfrac{\dfrac{T}{\dfrac{d}{2}}}{wl}[/tex]

[tex]\tau_{max}=\dfrac{2T}{dWl}[/tex]

[tex]240=\dfrac{2\times59.36}{2.159\times0.53975\times l}[/tex]

[tex]l=\dfrac{2\times59.36}{2.159\times0.53975\times240}[/tex]

[tex]l=0.4244\ m[/tex]

Hence, The length of the rectangular key is 0.4244 m

Answer:

The pressure of the air is 499.91 kPa.

Explanation:

Given that,

Initial pressure = 430 kPa

Temperature = 13.0+273=286 K

Final temperature = 59.5+273=332.5 K

We need to calculate the final pressure

Using relation of pressure and temperature

At constant volume,

[tex]\dfrac{P'}{P}=\dfrac{T'}{T}[/tex]

[tex]\dfrac{P'}{430}=\dfrac{332.5}{286}[/tex]

[tex]P'=\dfrac{332.5}{286}\times430[/tex]

[tex]P'=499.91\ kPa[/tex]

Hence,The pressure of the air is 499.91 kPa.

Answer:

Beats are the difference in frequency.

(a) is correct option.

Explanation:

Beat :

Beat is the difference of the frequency of two waves.

The difference in frequency is equal to the number of beat per second.

Amplitude :

Amplitude of the wave is the maximum displacement.

Frequency :

Frequency is the number oscillations of wave in per second.

Intensity :

Intensity is the power per unit area.

Hence, Beats are the difference in frequency.

Explanation:

Momentum is conserved.

a) In the first scenario, Olaf and the ball have the same final velocity.

mu = (M + m) v

(0.400 kg) (10.9 m/s) = (70.2 kg + 0.400 kg) v

v = 0.0618 m/s

b) In the second scenario, the ball has a final velocity of 8.10 m/s in the opposite direction.

mu = mv + MV

(0.400 kg) (10.9 m/s) = (0.400 kg) (-8.10 m/s) + (70.2 kg) v

v = 0.108 m/s

a) After Olaf catches the ball, Olaf and the ball will move at a speed of 0.062 m/s.

b) The speed of Olaf after the ball bounces off his chest is 0.11 m/s.    

a) We can find the speed of Olaf and the ball by conservation of linear momentum.

[tex] p_{i} = p_{f} [/tex]

[tex] m_{o}v_{i_{o}} + m_{b}v_{i_{b}} = m_{o}v_{f_{o}} + m_{b}v_{f_{b}} [/tex]

Where:

[tex] m_{o}[/tex]: is the mass of Olaf = 70.2 kg

[tex] m_{b}[/tex]: is the mass of the ball = 0.400 kg

[tex] v_{i_{o}}[/tex]: is the intial speed of Olaf = 0 (he is at rest)

[tex]v_{i_{b}}[/tex]: is the initial speed of the ball = 10.9 m/s

[tex] v_{f_{o}}[/tex] and [tex]v_{f_{b}} [/tex]: are the final speed of Olaf and the final speed of the ball, respetively.

Since Olaf catches the ball, we have that [tex] v_{f_{o}} = v_{f_{b}} = v [/tex], so:

[tex] m_{o}v_{i_{o}} + m_{b}v_{i_{b}} = v(m_{o} + m_{b}) [/tex]

We will take the direction of motion of the ball to the right side, and this will be the positive x-direction.  

By solving for "v" we have:

[tex] v = \frac{m_{o}v_{i_{o}} + m_{b}v_{i_{b}}}{m_{o} + m_{b}} = \frac{70.2 kg*0 + 0.400 kg*10.9 m/s}{70.2 kg + 0.400 kg} = 0.062 m/s [/tex]

Hence, Olaf and the ball will move at a speed of 0.062 m/s.

b) The final speed of Olaf after the collision can be calculated, again with conservation of linear momentum.

[tex] m_{o}v_{i_{o}} + m_{b}v_{i_{b}} = m_{o}v_{f_{o}} + m_{b}v_{f_{b}} [/tex]

In this case, since the ball hits Olaf and bounces off his chest, we have that [tex] v_{f_{o}} \neq v_{f_{b}}[/tex]

[tex] 0.400 kg*10.9 m/s = 70.2 kg*v_{f_{o}} + 0.400 kg*(-8.10 m/s) [/tex]      

The minus sign of the speed of the ball is because it moves to the negative direction of motion after the collision.  

[tex] v_{f_{o}} = \frac{0.400 kg*10.9 m/s + 0.400 kg*8.10 m/s}{70.2 kg} = 0.11 m/s [/tex]

Therefore, the speed of Olaf after the collision will be 0.11 m/s in the positive x-direction.                                  

You can see another example of conservation of linear momentum here: https://brainly.com/question/22257322?referrer=searchResults

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Answer:

Option C is the correct answer.

Explanation:

Considering vertical motion of ball:-

Initial velocity, u =  2 m/s

Acceleration , a = 9.81 m/s²

Displacement, s = 40 m

We have equation of motion s= ut + 0.5 at²

Substituting

   s= ut + 0.5 at²

    40 = 2 x t + 0.5 x 9.81 x t²

   4.9t²  + 2t - 40 = 0

   t = 2.66 s   or t = -3.06 s

So, time is 2.66 s.

Option C is the correct answer.

Answer:

4.78 x 10^-11 J

Explanation:

A = 1.5 x 10^-4 m^2

d = 2 mm = 2 x 10^-3 m

V = 12 V

Let C be the capacitance of the capacitor

C = ε0 A / d

C = (8.854 x 10^-12 x 1.5 x 10^-4) / (2 x 10^-3)

C = 6.64 x 10^-13 F

Energy stored, U = 1/2 CV^2

U = 0.5 x 6.64 x 10^-13 x 12 x 12

U = 4.78 x 10^-11 J

The energy stored in a parallel-plate capacitor connected to a 12-V battery, with plate area of 1.5 x 10^-4 m^2 and separation of 2.0 mm, is calculated using the formula U = (1/2)CV^2 and is found to be 4.8 x 10^-11 J.

The capacitance C of a parallel-plate capacitor is given by the formula C = ε_0 * A / d, where ε_0 is the vacuum permittivity (ε_0 = 8.85 x 10^-12 F/m), A is the area of the plates, and d is the separation between the plates.

Given that the area A is 1.5 x 10^-4 m^2, the separation d is 2.0 mm = 2.0 x 10^-3 m, and the voltage V is 12 V, we can plug in these values to first determine capacitance and then calculate the energy stored.

First, calculate the capacitance:

C = ε_0 * A / d = (8.85 x 10^-12 F/m)(1.5 x 10^-4 m^2) / (2.0 x 10^-3 m) = 6.6 x 10^-12 F

Next, calculate the energy stored:

U = (1/2)CV^2 = (1/2)(6.6 x 10^-12 F)(12 V)^2 = 4.8 x 10^-11 J

Therefore, the energy stored in the capacitor is 4.8 x 10^-11 J, which corresponds to option (a).

Answer:

The water will flow at a speed of 3,884 m/s

Explanation:

Torricelli's equation

v = [tex]\sqrt{2gh}[/tex]

*v = liquid velocity at the exit of the hole

g = gravity acceleration

h = distance from the surface of the liquid to the center of the hole.

v = [tex]\sqrt{2*9,8m/s^2*0,77m}[/tex] = 3,884 m/s

We have that for the Question"A large container, 120 cm deep is filled with water. If a small hole is punched in its side 77.0 cm from the top, at what initial speed will the water flow from the hole" it can be said that initial speed  the water flow from the hole is

From the question we are told

A large container, 120 cm deep is filled with water. If a small hole is punched in its side 77.0 cm from the top, at what initial speed will the water flow from the hole? Please use a equation and explain every step

Generally the equation for the water flow speed   is mathematically given as

[tex]v=\sqrt(2gh)\\\\v=\sqrt{2*9.8*0.77}[/tex]

v=3.88m/s

Therefore

initial speed  the water flow from the hole is

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Answer:

final equilibrium temperature of the system is ZERO degree Celcius

Explanation:

Hear heat given by water + iron = heat absorbed by ice

so here first we will calculate the heat given by water + iron

[tex]Q_1 = m_1s_2\Delta T_1 + m_2 s_2 \Delta T_1[/tex]

[tex]Q_1 = (200)(0.450)(40 - T) + (50)(4.186)(40 - T)[/tex]

now the heat absorbed by ice so that it will melt and come to the final temperature

[tex]Q_2 = m s \Delta T + mL + m s_{water}\Delta T'[/tex]

[tex]Q_2 = 50(2.09)(0 + 6) + 50(335) + 50(4.186)(T - 0)[/tex]

now we will have

[tex]17377 + 209.3T = 3600 - 90T + 8372 - 209.3T[/tex]

[tex]17377 + 209.3T + 90T + 209.3T = 11972[/tex]

[tex]T = -10.6[/tex]

since it is coming out negative which is not possible so here the ice will not completely melt

so final equilibrium temperature of the system is ZERO degree Celcius

Answer:

[tex]\omega' = 0.89\omega[/tex]

Explanation:

Rotational inertia of uniform solid sphere is given as

[tex]I = \frac{2}{5}MR^2[/tex]

now we have its angular speed given as

angular speed = [tex]\omega[/tex]

now we have its final rotational kinetic energy as

[tex]KE = \frac{1}{2}(\frac{2}{5}MR^2)\omega^2[/tex]

now the rotational inertia of solid cylinder about its axis is given by

[tex]I = \frac{1}{2}MR^2[/tex]

now let say its angular speed is given as

angular speed = [tex]\omega'[/tex]

now its rotational kinetic energy is given by

[tex]KE = \frac{1}{2}(\frac{1}{2}MR^2)\omega'^2[/tex]

now if rotational kinetic energy of solid sphere is same as rotational kinetic energy of solid sphere then

[tex]\frac{1}{2}(\frac{2}{5}MR^2)\omega^2 = \frac{1}{2}(\frac{1}{2}MR^2)\omega'^2[/tex]

[tex]\frac{2}{5}\omega^2 = \frac{1}{2}\omega'^2[/tex]

[tex]\omega' = 0.89\omega[/tex]

Answer:

w_cyl = ±√(4/5) ω

Explanation:

Kinetic energy

E = (1/2)Iw²

where I is the moment of inertia and w the angular frequency of rotation.

The moment of inertia of a solid sphere of mass M and radius R is:

I = (2/5)MR²,  

Solid cylinder is of mass M and radius R

I = (1/2)MR²

Equate the energies through

(1/2)×(2/5) M R²× (w_sphere)² = (1/2)× (1/2) MR² × (w_cyl)²

(w_cyl)² = (4/5)(w_sphere)²

w_cyl = ±√(4/5) ω

The energy of rotation is independent of the direction of rotation

Answer:

Final velocity is 181.61 m/s at angle 40.44° below horizontal.

Explanation:

Initial horizontal velocity = 170 cos 35.6 = 138.23 m/s

Final horizontal velocity = 138.23 m/s

Considering vertical motion of projectile:

Initial vertical velocity, u = 170 sin 35.6 = 98.96 m/s

Acceleration, a = -9.81 m/s²

Displacement, s = -208 m

We have v² = u² + 2as

Substituting

             v² = 98.96² + 2 x -9.81 x -208

             v = 117.79 m/s

Final velocity,

            [tex]v=\sqrt{138.23^2+117.79^2}=181.61m/s[/tex]

            [tex]\theta =tan^{-1}\left ( \frac{117.79}{138.23}\right )=40.44^0[/tex]

Final velocity is 181.61 m/s at angle 40.44° below horizontal.