A pair of eyeglass frames are made of an epoxy plastic (coefficient of linear expansion = 1.30 ✕ 10−4°C−1). At room temperature (20.0°C), the frames have circular lens holes 2.34 cm in radius. To what temperature must the frames be heated if lenses 2.35 cm in radius are to be inserted into them? °C

Answer:

1.90 x 10² cal

1.90 x 10² cal

0.133 cal/(g °C)

Explanation:

For water :

[tex]m_{w}[/tex] = mass of water = 112 g

[tex]c_{w}[/tex] = specific heat of water = 1 cal/(g °C)

[tex]T_{wi}[/tex] = initial temperature of water = 90.0 °C

[tex]T_{wf}[/tex] = final temperature of water = 88.3 °C

[tex]Q_{w}[/tex] = Heat lost by water

Heat lost by water is given as

[tex]Q_{w}= m_{w}c_{w}(T_{wi} - T_{wf})[/tex]

[tex]Q_{w}[/tex] = (112) (1) (90.0 - 88.3)

[tex]Q_{w}[/tex] = 1.90 x 10² cal

[tex]Q_{B}[/tex] = Heat gained by the block

As per conservation of energy

Heat gained by the block = Heat lost by water

[tex]Q_{B}[/tex] = [tex]Q_{w}[/tex]

[tex]Q_{B}[/tex] = 1.90 x 10² cal

For Block :

[tex]m_{B}[/tex] = mass of block = 21.0 g

[tex]c_{B}[/tex] = specific heat of block

[tex]T_{bi}[/tex] = initial temperature of block = 20.0 °C

[tex]T_{bf}[/tex] = final temperature of block = 88.3 °C

[tex]Q_{B}[/tex] = Heat gained by Block = 1.90 x 10² cal

Heat gained by water is given as

[tex]Q_{B}[/tex] = m_{B}c_{B}(T_{bf} - T_{bi})[/tex]

1.90 x 10²  = (21.0) (88.3 - 20.0) c_{B}

c_{B} = 0.133 cal/(g °C)

Answer:

Thickness of magazine = 9.42 mm.

Explanation:

Considering vertical motion of spider:-

Initial velocity, u =  0.870 sin 35 = 0.5 m/s

Acceleration , a = -9.81 m/s²

Time, t = 0.077 s

We have equation of motion s= ut + 0.5 at²

Substituting

   s= 0.5 x 0.077 - 0.5 x 9.81 x 0.077²

    s = 9.42 x 10⁻³ m = 9.42 mm

Thickness of magazine = 9.42 mm.

The thickness of the magazine is determined using the vertical component of the spider's initial velocity and kinematic equations for constant acceleration. By calculating and converting the vertical displacement to millimeters, we can find the thickness of the magazine the spider lands on.

Given that the initial velocity in the vertical direction (Vy) can be found using the formula Vy = V * sin(θ), where V is the initial velocity of the spider and θ is the angle of projection, we can calculate Vy = (0.870 m/s) * sin(35.0°).

The vertical displacement (y) can then be calculated using the kinematic equation for constant acceleration, y = Vy * t + (1/2) * g * t^2, where t is the time of flight, and g is the acceleration due to gravity (approximately -9.81 m/s^2). Since we want to find the displacement in the upward direction and the spider lands on the magazine after 0.0770 s, we are looking for the magnitude of y when the spider lands.

Substituting the values we have: y = (0.870 m/s) * sin(35.0°) * 0.0770 s + (0.5) * (-9.81 m/s^2) * (0.0770 s)^2. The negative sign in the acceleration term accounts for the direction of gravity, which is downwards. After calculating this expression, we convert the result from meters to millimeters (by multiplying by 1000) to obtain the thickness of the magazine.

Answer:

The formula is Work = Force x distance moved

so that would be 0.600 x 5 = 3 Joules which is 0.003 kilojoules

The work done in by the attendant on a can of soup is 3 joules or 0.00072 Kilocalories

Work in physics is defined as the dot product of force and the displacement produced by it.

Given is an attendant who pushed 0.600 m horizontally with a force of 5.00 N.

From the definition of work done, we can write -

W = F.d

W = Fd cosФ

Then angle between force and displacement is 0°. Therefore, cosФ is equal to 1.

W = Fd

W = 5 x 0.6

W = 3 joules = 3/4184 = 0.00072 Kilocalories

Therefore, the work done in by the attendant on a can of soup is 3 joules or 0.00072 Kilocalories

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To calculate the temperature difference between the inner and outer surfaces of the glass bulb in a 100 W bulb that generates 95W of heat, we use a formula from the principles of heat conduction where we input parameters including the heat generated by the bulb, the thermal conductivity of the glass, the surface area of the glass and the thickness of the glass.

In order to calculate the temperature difference between the inner and outer surfaces of the glass bulb, we need to use the formula for heat conduction, which is given by the formula Q = (k*A*ΔT)/d, where Q is the heat generated by the bulb, k is the thermal conductivity of the glass, A is the surface area of the glass, ΔT is the temperature difference, and d is the thickness of the glass. In this case, we know that the light bulb generates 95W of heat, the radius of the bulb is 3.0 cm, the thickness of the glass is 0.50 mm. Assuming the thermal conductivity of the glass to be 0.8 W/m.K, we can substitute these values into the formula to calculate the temperature difference ΔT = Qd / (k*A). Note that here, the surface area of the glass, A = 4πr².

As you can see, the calculation requires a clear grasp of the concepts of heat conduction, thermal conductivity, and physical constants of materials. Understanding how these factors interact is key to solving problems about heat transfer in Physics.

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Answer:

Heat energy needed = 3036.17 kJ

Explanation:

We have

     heat of fusion of water = 334 J/g

     heat of vaporization of water = 2257 J/g

     specific heat of ice = 2.09 J/g·°C

     specific heat of water = 4.18 J/g·°C

     specific heat of steam = 2.09 J/g·°C

Here wee need to convert 1 kg ice from -13°C to vapor at 100°C

First the ice changes to -13°C from 0°C , then it changes to water, then its temperature increases from 0°C to 100°C, then it changes to steam.

Mass of water = 1000 g

Heat energy required to change ice temperature from -13°C to 0°C

          H₁ = mcΔT = 1000 x 2.09 x 13 = 27.17 kJ

Heat energy required to change ice from 0°C to water at 0°C

          H₂ = mL = 1000 x 334 = 334 kJ

Heat energy required to change water temperature from 0°C to 100°C  

          H₃ = mcΔT = 1000 x 4.18 x 100 = 418 kJ    

Heat energy required to change water from 100°C to steam at 100°C  

          H₄ = mL = 1000 x 2257 = 2257 kJ    

Total heat energy required

          H = H₁ +  H₂ + H₃ + H₄ = 27.17 + 334 + 418 +2257 = 3036.17 kJ

Heat energy needed = 3036.17 kJ

Answer:

Part a)

L = 25.13 kg m^2/s

Part b)

N = 3200 rev

Part c)

torque = 0.837 Nm

Explanation:

Part a)

As we know that angular frequency is given as

[tex]\omega = 2\pi f[/tex]

[tex]\omega = 2\pi(800/60)[/tex]

[tex]\omega = 83.77 rad/s[/tex]

now the angular momentum is given as

[tex]L = I\omega[/tex]

[tex]L = (0.3)(83.77)[/tex]

[tex]L = 25.13 kg m^2/s[/tex]

Part b)

angular acceleration of fan is given as

[tex]\alpha = 2\pi\frac{800/60}{30}[/tex]

[tex]\alpha = 2.79 rad/s^2[/tex]

total number of revolutions are

[tex]N = \frac{1}{4\pi}\alpha t^2[/tex]

[tex]N = \frac{1}{4\pi}(2.79)(120^2)[/tex]

[tex]N = 3200 rev[/tex]

Part c)

Torque is given as

[tex]\tau = I\alpha[/tex]

[tex]\tau = (0.30)(2.79) = 0.837 Nm[/tex]

Answer:

1.53 x 10^-5 m^2

Explanation:

use the Stefan's law

Energy per unit time = σ x A x T^4

σ = 5.67 x 10 -8 W/m^2 K^4

96 = 5.67 x 10^-8 x A x (3242)^4

A = 1.53 x 10^-5 m^2

Answer:

a) 1/4 what it is now

Explanation:

As we know that force of gravitation between two planets at some distance "r" from each other is given as

[tex]F_g = \frac{Gm_1m_2}{r^2}[/tex]

now since we know that if the distance between Earth and Sun is changed

So the force of gravity will be given as

[tex]\frac{F_g'}{F_g} = \frac{r_1^2}{r_2^2}[/tex]

now we know that the distance between sun and earth is changed to twice the initial distance between them

so we have

[tex]r_2 = 2r_1[/tex]

so new gravitational force between sun and earth is given as

[tex]F_g' = \frac{r_1^2}{(2r_1)^2}F_g[/tex]

[tex]F_g' = \frac{1}{4}F_g[/tex]

Answer:

v = 11.3 m/s

Explanation:

Since there is no external force on the system of two cars

so here momentum of the system of two cars will remain constant

it is given here that two cars combined and move together at 41.5 degree North of East after collision

so here direction of final momentum is given as

[tex]tan\theta = \frac{P_y}{P_x}[/tex]

now we have initial momentum in the same direction

[tex]P_1 = m(10 m/s)\hat j[/tex]

[tex]P_2 = mv\hat i[/tex]

now we have

[tex]\frac{P_1}{P_2} = tan\theta[/tex]

[tex]\frac{m(10 m/s)}{mv} = tan41.5[/tex]

[tex]\frac{10}{v} = 0.88[/tex]

[tex]v = 11.3 m/s[/tex]

Using the principles of conservation of momentum and trigonometry, the initial velocity of car B can be calculated to be approximately 9.75 m/s.

The problem can be solved using the principles of conservation of momentum. In a two-dimensional collision, the components of the momentum in the X and Y directions are separately conserved. Since we're given that the cars have equal mass and they stick together after colliding to make a single object, the magnitude of the velocity of the combined object after the collision must be equal to the vector sum of their initial velocities.

Car A is moving northward and thus its velocity is contributing only to the Y component of the total momentum. Car B, on the other hand, is moving eastward, so its velocity is contributing to the X component. When the cars collide and stick together, they move at an angle of 41.5° north of east, meaning there is both an X and Y component to their velocity.

Using trigonometry, the X component (eastward direction, i.e., car B's direction) can be found by VB = VA * tan(Θ). Given that VA is 10 m/s and Θ is 41.5 degrees, VB can be calculated to be approximately 9.75 m/s.

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Explanation:

It is given that,

Diameter of the copper wire, d = 2.3 cm

Radius of copper wire, r = 1.15 cm = 0.0115 m

Distance between bird's feet, l = 3.9 cm = 0.039 m

The resistivity of the wire, [tex]\rho=1.7\times 10^{-8}\ \Omega-m[/tex]

We need to find the potential difference between bird's feet. The resistance of the wire is calculated by :

[tex]R=\rho\times \dfrac{l}{A}[/tex]

[tex]R=1.7\times 10^{-8}\ \Omega-m\times \dfrac{0.039\ m}{\pi(0.0115\ m)^2}[/tex]

R = 0.00000159 ohms

[tex]R=1.59\times 10^{-6}\ \Omega[/tex]

Let V is the potential difference between bird's feet. It can be calculated using Ohm's law as :

[tex]V=I\times R[/tex]

[tex]V=45\ A\times 1.59\times 10^{-6}\ \Omega[/tex]

V = 0.000071 volts

or

[tex]V=7.1\times 10^{-5}\ volts[/tex]

So, the potential difference between the bird's feet is [tex]7.1\times 10^{-5}\ volts[/tex]. Hence, this is the required solution.

Even with a large estimated resistance for the bird (say, R_bird = 1000 Ω), the potential difference (V_bird) would be very small (V_bird = 45 A * 1000 Ω ≈ 45000 V).  However, in reality, most of the current will flow through the much lower resistance wire, making the actual potential difference between the bird's feet much closer to zero.

Calculation (for illustration purposes only):

Even though the potential difference is negligible, we can estimate its upper bound (worst-case scenario) by assuming all the current flows through the bird's body.

Wire Resistance:

Diameter (d) = 2.3 cm = 0.023 m

Radius (r) = d/2 = 0.0115 m

Wire length between feet (l) = 3.9 cm = 0.039 m

Resistivity (ρ) = 1.7 x 10^-8 Ω · m

Wire Resistance (R_wire) = ρ * (l / π * r^2) = (1.7 x 10^-8 Ω · m) * (0.039 m / (π * (0.0115 m)^2)) ≈ 4.2 x 10^-7 Ω (very small)

Assuming All Current Through Bird:

Current (I) = 45 A

Bird's Body Resistance (R_bird) = V_bird / I (Since we don't have V_bird, this is just a placeholder)

Total Circuit Resistance (ignoring wire resistance):

R_total = R_bird + R_wire ≈ R_bird (because R_wire is negligible)

Potential Difference Across Bird (upper bound):

V_bird = I * R_total = I * R_bird (since R_wire is negligible)

Answer:

The acceleration and time are 1.95 m/s and 12.5 s.

Explanation:

Given that,

Speed = 80 ft/s =24.384 m/s

Distance = 500 ft =152.4 m

We need to calculate the acceleration

Using third equation of motion

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

[tex]a = \dfrac{v^2-u^2}{2s}[/tex]

Where, u = initial velocity

v = final velocity

a = acceleration

s = distance

Put the value in the equation

[tex]a=\dfrac{(24.384)^2-0}{2\times152.4}[/tex]

[tex]a=1.95\ m/s^2[/tex]

We need to calculate the time

Using first equation of motion

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

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

[tex]t=\dfrac{24.384-0}{1.95}[/tex]

[tex]t =12.5\ s[/tex]

Hence, The acceleration and time are 1.95 m/s and 12.5 s.

q = 1156363.6W/m².

To calculate the heat flux per unit area (W/m²) of a sheet made of metal:

q = -k(ΔT/Δx)

q = -k[(T₂ - T₁)/Δx]

Where, k is the thermal conductivity of the metal, ΔT is the temperature difference and Δx is the thick.

With Δx = 11 mm = 11x10⁻³m, T₂ = 350°C and T₁ = 110°C, and k = 53.0 W/m-K:

q = -53.0W/m-K[(110°C - 350°C)/11x10⁻³m

q = 1156363.6W/m²

Answer:

The acceleration and charge are 17.122 m/s² and [tex]8.6\times10^{-5}\ C[/tex]

Explanation:

Given that,

Mass of object = 0.8 g

Electric field = 159 N/C

Distance = 72 m

Time = 2.9 s

We know that,

The electric force is

[tex]F = Eq[/tex]....(I)

The newton's second law

[tex]F=ma[/tex]

Put the value of F in the equation (I)

[tex]ma=Eq[/tex]...(II)

We calculate the acceleration

Using equation of motion

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

[tex]a =\dfrac{2s}{t^2}[/tex]

[tex]a=\dfrac{2\times72}{(2.9)^2}[/tex]

[tex]a=17.122\ m/s^2[/tex]

From equation (II)

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

[tex]q=\dfrac{0.8\times10^{-3}\times17.122}{159}[/tex]

[tex]q=0.000086148427673\ C[/tex]

[tex]q=8.6\times10^{-5}\ C[/tex]

Hence, The acceleration and charge are 17.122 m/s² and [tex]8.6\times10^{-5}\ C[/tex]

Answer:

angular speed of both the children will be same

Explanation:

Rate of revolution of the merry go round is given as

f = 4.04 rev/min

so here we have

[tex]f = \frac{4.04}{60} =0.067 rev/s[/tex]

here we know that angular frequency is given as

[tex]\omega = 2\pi f[/tex]

[tex]\omega = 2\pi(0.067)[/tex]

[tex]\omega = 0.42 rad/s[/tex]

now this is the angular speed of the disc and this speed will remain same for all points lying on the disc

Angular speed do not depends on the distance from the center but it will be same for all positions of the disc

Answer:

f = 91.1 Hz

Explanation:

As we know that it is given here that wavelength of sound in platinum is same as the wavelength of sound in air

so we can use the formula of wavelength in two mediums

[tex]\frac{speed}{frequency} = \frac{speed}{frequency}[/tex]

now it is given that

[tex]v_{air} = 340 m/s[/tex]

[tex]v_{pt} = 2800 m/s[/tex]

frequency of sound in platinum is 750 Hz

now frequency of sound in air = f

now from above formula

[tex]\frac{340}{f} = \frac{2800}{750}[/tex]

[tex]f = 91.1 Hz[/tex]

Answer:

a)  Initial angular speed = 30 rad/s

b) Final angular speed = 70 rad/s        

Explanation:

a) We have equation of motion s = ut + 0.5at²

    Here s = 400 radians

              t = 8 s

              a = 5 rad/s²

    Substituting

             400 = u x 8 + 0.5 x 5 x 8²

              u = 30 rad/s

   Initial angular speed = 30 rad/s

b) We have equation of motion v = u + at

     Here u = 30 rad/s

               t = 8 s

              a = 5 rad/s²  

    Substituting

             v = 30 + 5 x 8 = 70 rad/s    

   Final angular speed = 70 rad/s        

Final answer:

The self-inductance of the coil is calculated using the energy stored in its magnetic field and the current flowing through it. By rearranging the formula W = (1/2) * L * I², the self-inductance L is found to be 0.11875 H.

Explanation:

To calculate the self-inductance of the coil, we can use the formula for the energy stored in the magnetic field of an inductor:

Energy (W) = (1/2) * L * I²

Where:

W is the energy stored in the magnetic field

L is the self-inductance of the coil

I is the current flowing through the coil

From the question, we're given that the energy (W) is 9.5 x 10⁻³ J, and the current (I) is 0.40 A. We can rearrange the formula to solve for L:

L = 2W / I²

Substitute the given values:

L = 2 * (9.5 x 10⁻³ J) / (0.40 A)²

L = 2 * (9.5 x 10⁻³ J) / (0.16 A²)

L = (19 x 10⁻³ J) / (0.16 A²)

L = 0.11875 H

Therefore, the self-inductance of the coil is 0.11875 H (henrys).

Answer: The freezing point of solution is -3.34°C

Explanation:

Vant hoff factor for ionic solute is the number of ions that are present in a solution. The equation for the ionization of calcium nitrate follows:

[tex]Ca(NO_3)_2(aq.)\rightarrow Ca^{2+}(aq.)+2NO_3^-(aq.)[/tex]

The total number of ions present in the solution are 3.

[tex]Molality=\frac{m_{solute}\times 1000}{M_{solute}\times W_{solvent}\text{ in grams}}[/tex]

Where,

[tex]m_{solute}[/tex] = Given mass of solute [tex](Ca(NO_3)_2)[/tex] = 11.3 g

[tex]M_{solute}[/tex] = Molar mass of solute [tex](Ca(NO_3)_2)[/tex] = 164  g/mol

[tex]W_{solvent}[/tex] = Mass of solvent (water) = 115 g

Putting values in above equation, we get:

[tex]\text{Molality of }Ca(NO_3)_2=\frac{11.3\times 1000}{164\times 115}\\\\\text{Molality of }Ca(NO_3)_2=0.599m[/tex]

[tex]\Delta T=iK_fm[/tex]

where,

i = Vant hoff factor = 3

[tex]K_f[/tex] = molal freezing point depression constant = 1.86°C/m.g

m = molality of solution = 0.599 m

Putting values in above equation, we get:

[tex]\Delta T=3\times 1.86^oC/m.g\times 0.599m\\\\\Delta T=3.34^oC[/tex]

Depression in freezing point is defined as the difference in the freezing point of water and freezing point of solution.

[tex]\Delta T=\text{freezing point of water}-\text{freezing point of solution}[/tex]

[tex]\Delta T[/tex] = 3.34 °C

Freezing point of water = 0°C

Freezing point of solution = ?

Putting values in above equation, we get:

[tex]3.34^oC=0^oC-\text{Freezing point of solution}\\\\\text{Freezing point of solution}=-3.34^oC[/tex]

Hence, the freezing point of solution is -3.34°C

Final answer:

The freezing point of a solution made by dissolving 11.3 g of Ca(NO3)2 in 115 g of water is -3.34 °C. To find this, we calculate molality, account for the dissociation of ions and use the freezing point depression constant for water.

Explanation:

To calculate the freezing point depression of a solution of Ca(NO3)2 in water, we first determine the molality of the solution. With the provided mass of Ca(NO3)2 (11.3 g) and its formula weight (164 g/mol), we find there are 0.0689 moles of Ca(NO3)2. Since we only have 115 g of water, to convert to kilograms, we have 0.115 kg. The molality (m) is then 0.0689 moles / 0.115 kg = 0.599 m. Since Ca(NO3)2 dissociates into three ions (Ca2+, 2NO3-), the van't Hoff factor (i) is 3.

The depression of the freezing point is determined using the formula ΔTf = i * Kf * m, where Kf is the molal freezing point depression constant for water (1.86 °C/m). So the depression is ΔTf = 3 * 1.86 °C/m * 0.599 m = 3.34 °C.

The freezing point of the solution is then 0 °C - 3.34 °C = -3.34 °C, which is the answer.

Answer:

22.89 N

Explanation:

F = 37 N

Let the acceleration in the system is a and f be the force between the tewo blocks.

Apply the Newton's second law

By the free body diagrams

F - f = 7.4 x a .... (1)

f = 12 x a ...... (2)

Adding both of them

37 = 19.4 x a

a = 1.9 m/s^2

Put in equation (2)

f = 12 x 1.9 = 22.89 N

Answer:

Required charge [tex]q=2.6\times 10^{9}C[/tex].

[tex]n=1.622\times 10^{10}\ electrons[/tex]

Explanation:

Given:

Diameter of the isolated plastic sphere = 25.0 cm

Magnitude of the Electric field = 1500 N/C

now

Electric field (E) is given as:

[tex]E =\frac{kq}{r^2}[/tex]

where,

k = coulomb's constant = 9 × 10⁹ N

q = required charge

r = distance of the point from the charge where electric field is being measured

The value of r at the just outside of the sphere = [tex]\frac{25.0}{2}=12.5cm=0.125m[/tex]

thus, according to the given data

[tex]1500N/C=\frac{9\times 10^{9}N\times q}{(0.125m)^2}[/tex]

or

[tex]q=\frac{0.125^2\times 1500}{9\times 10^{9}}[/tex]

or

Required charge [tex]q=2.6\times 10^{9}C[/tex].

Now,

the number of electrons (n) required will be

[tex]n=\frac{required\ charge}{charge\ of\ electron}[/tex]

or

[tex]n=\frac{2.6\times 10^{-9}}{1.602\times 10^{-19}}[/tex]

or

[tex]n=1.622\times 10^{10}\ electrons[/tex]

To produce an electric field of 1500 N/C just outside the surface of the plastic sphere, approximately 337.5 nC of excess charge must be distributed uniformly within its volume.

To produce an electric field just outside the surface of the sphere, the excess charge on the surface of the sphere should be uniformly distributed within its volume. The electric field just outside a uniformly charged sphere is given by:

E = k * q / r2

Where E is the electric field, k is the Coulomb constant [tex](9 * 109 N m2/C^2)[/tex] q is the charge, and r is the distance from the center of the sphere to the point outside the surface. In this case, E is given as 1500 N/C and r is half the diameter of the sphere (12.5 cm).

Substituting the values into the equation, we can solve for q:

[tex]1500 N/C = (9 * 109 N m2/C2) * q / (0.125 m)^2[/tex]

Solving for q, we find that the excess charge on the surface of the sphere should be approximately [tex]3.375 x 10-7 C,[/tex] or about 337.5 nC.

Explanation:

The velocity field o a steady flow is given by :

[tex]V=(U_o+bx)i[/tex]

Where

U₀ and b are constant

We know that, the acceleration is given by :

[tex]a=\dfrac{dV}{dt}[/tex]

[tex]a=\dfrac{d((U_o+bx)i)}{dt}[/tex]

a = b i

a = 0.3 m/s²

So, acceleration corresponding to this velocity is 0.3 m/s². Hence, this is the required solution.

Answer:

147,099,713.4 km

152,096,286.6 km

Explanation:

a = 149598000 km

e = 0.0167

The formula to find the perihelion

Rp = a ( 1 - e) = 149598000 ( 1 - 0.0167) = 147,099,713.4 km

The formula for aphelion

Ra = a ( 1 + e) = 149598000 ( 1 + 0.0167) = 152,096,286.6 km

To find the minimum and maximum distances from Earth to the Sun (perihelion and aphelion), we calculate them using Earth's semi-major axis of 149,598,000 kilometers and the eccentricity of 0.0167. The perihelion is 147,099,014 kilometers, and the aphelion is 152,096,986 kilometers.

The student's question revolves around finding the minimum (perihelion) and maximum (aphelion) distances from the Earth to the Sun, given the length of half of the major axis — also known as the semi-major axis — and the eccentricity of Earth's orbit. The semi-major axis (a) is 149,598,000 kilometers and the eccentricity (e) is 0.0167. The distance from the center of the ellipse, where Earth's orbit is, to the focus (c) is equal to the product of the semi-major axis and the eccentricity (c = ae).

The perihelion distance is the semi-major axis minus the distance c, resulting from the Earth being at the closest point to the Sun in its orbit. Conversely, the aphelion distance is the semi-major axis plus the distance c, when Earth is farthest from the Sun. Therefore, the perihelion (rp) can be calculated as rp = a - c, and the aphelion (ra) as ra = a + c.

Using the formula c = ae, we find that c is approximately 2,498,986 kilometers (149,598,000 km * 0.0167). Thus:

Answer:

26.8 m/s

Explanation:

[tex]v[/tex]  = constant speed of the car

[tex]V[/tex]  = speed of sound = 343 m/s

[tex]f[/tex] = actual frequency of the horn

[tex]f_{app}[/tex] = frequency heard as the car approach = 76 Hz

frequency heard as the car approach is given as

[tex]f_{app}=\frac{vf}{V - v}[/tex]

[tex]76 =\frac{vf}{343 - v}[/tex]                               eq-1

[tex]f_{rec}[/tex] = frequency heard as the car recedes = 65 Hz

frequency heard as the car goes away is given as

[tex]f_{rec}=\frac{vf}{V + v}[/tex]

[tex]65 =\frac{vf}{343 + v}[/tex]                                  eq-2

dividing eq-1 by eq-2

[tex]\frac{76}{65}=\frac{343+v}{343-v}[/tex]

[tex]v[/tex] = 26.8 m/s

Final answer:

To determine the car's speed using the Doppler Effect, we calculate the difference in observed sound frequencies as the car approaches and moves away. Applying formulas for Doppler Effect calculations, the speed of the car comes out to be approximately 14.6 m/s.

Explanation:

The question revolves around the phenomenon known as the Doppler Effect, which is observed when a sound source moves relative to an observer. To calculate the speed v of the car, we use the Doppler Effect equations for sound frequencies heard when the source is moving towards and then away from the observer:

For the source approaching:

f' = f * ((v + vo) / (v - vs))

, where:

f' is the observed frequency when the source is approaching (76 Hz)

f is the original frequency emitted by the source

v is the speed of sound (343 m/s)

vo is the speed of the observer (0 m/s, since the observer is stationary)

vs is the speed of the source (the car's speed, what we are solving for)

For the source receding:

f'' = f * ((v - vo) / (v + vs))

, where:

f'' is the observed frequency when the source is receding (65 Hz)

To find the car's speed, we need to solve for vs in both equations. By eliminating f (since it's the same for both equations), we can solve for vs. Using these equations, we determine that the speed of the car is approximately 14.6 m/s.

Answer:

0.686 Ohm

Explanation:

Heat energy H = 50 cal = 50 x 4.2 J = 210 J, time t = 1 second, V = 12 V

Let R be the resistance.

Heat energy = V^2 x t / R

210 = 12 x 12 x 1 / R

210 = 144 / R

R = 144 / 210 = 0.686 Ohm

The resistance of the heater wire is approximately 0.688 ohms.

To calculate the resistance of the heater wire, we need to use the information that 50 cal of heat is generated per second when a potential difference of 12 V is applied across the leads. The first step is to convert the heat from calories to joules, which is the standard unit of energy in Physics. Since 1 calorie is equivalent to 4.184 joules, we can calculate the power (P) in joules per second (or watts) by multiplying the heat generated per second by this conversion factor:

P = 50 cal/s ×4.184 J/cal = 209.2 J/s = 209.2 W

Now, we can use the formula for electric power P = V^2/R, where P is the power, V is the potential difference, and R is the resistance. By rearranging the formula to solve for R, we get R = V^2/P.

Plugging our values into this equation gives us:

R = (12 V)^2 / (209.2 W) = 144 V^2 / 209.2 W = approx 0.688 ohms.

Answer:

The time required for the package to hit the ground = 14.63 seconds.

Explanation:

Considering vertical motion of care package:-

Initial velocity, u =  0 m/s

Acceleration , a = 9.81 m/s²

Displacement, s = 1050 m

We have equation of motion s= ut + 0.5 at²

Substituting

   s= ut + 0.5 at²

    1050 = 0 x t + 0.5 x 9.81 x t²

    t = 14.63 seconds

The time required for the package to hit the ground = 14.63 seconds.

To determine the time required for the care package to hit the ground, we can separate the vertical and horizontal components of the motion. By using the equation h = (1/2)gt^2 and plugging in the values h = 1050 m and g = 9.8 m/s^2, we find that it will take approximately 14.63 seconds for the care package to hit the ground.

To determine the time required for the care package to hit the ground, we can separate the vertical and horizontal components of the motion. Since the horizontal velocity of the airplane does not change, it will not affect the time of flight. The vertical motion can be analyzed using the equation of motion for free fall, which is h = (1/2)gt^2, where h is the initial vertical displacement, g is the acceleration due to gravity, and t is the time of flight. In this case, the initial vertical displacement is 1050 meters and the acceleration due to gravity is approximately 9.8 m/s^2. Plugging these values into the equation, we can solve for t and find the time required for the care package to hit the ground.

By using the equation h = (1/2)gt^2 and plugging in the values h = 1050 m and g = 9.8 m/s^2, we have 1050 = (1/2)(9.8)t^2. Solving for t, we get t^2 = (1050) / (0.5)(9.8) = 214.29. Taking the square root of both sides, we find t ≈ 14.63 seconds. Therefore, it will take approximately 14.63 seconds for the care package to hit the ground.

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Answer:

(c) 3P/5

Explanation:

The formula to calculate the power is:

[tex]P=\frac{W}{T}[/tex]

where

W is the work done

T is the time required for the work to be done

In the second part of the problem, we have

Work done: 3W

Time interval: 5T

So the power required is

[tex]P=\frac{3W}{5T}=\frac{3}{5}\frac{W}{T}=\frac{3}{5}P[/tex]

Answer:

at the highest point of the path the acceleration of ball is same as acceleration due to gravity

Explanation:

At the highest point of the path of the ball the speed of the ball becomes zero as the acceleration due to gravity will decelerate the motion of ball due to which the speed of ball will keep on decreasing and finally it comes to rest

So here we will say that at the highest point of the path the speed of the ball comes to zero

now by the force diagram we can say that net force on the ball due to gravity is given by

[tex]F_g = mg[/tex]

now the acceleration of ball is given as

[tex]a = \frac{F_g}{m}[/tex]

[tex]a = \frac{mg}{m} = g[/tex]

so at the highest point of the path the acceleration of ball is same as acceleration due to gravity

Final answer:

The magnitude of the acceleration of a tennis ball at its highest point, in an evacuated chamber and under the influence of gravity alone, is 9.8 m/s². This remains constant regardless of the ball's position in its trajectory.

Explanation:

The question asks about the magnitude of the acceleration of a tennis ball at its highest point when it is shot vertically upward in an evacuated chamber with an initial speed. This scenario is governed by the principles of classical mechanics, specifically under the domain of gravity's influence on objects in motion. Regardless of whether the object is moving up or down or is at its peak height, the acceleration due to gravity on Earth remains constant.

Acceleration at the highest point, or at any point during its motion under gravity alone (assuming no air resistance), is -9.8 m/s2. It's essential to understand that the negative sign indicates the direction of acceleration, which is towards the center of the Earth but does not affect the magnitude of acceleration. Hence, the magnitude of acceleration is 9.8 m/s2, indicating that gravity is the only force acting on the ball throughout its trajectory.

Answer:

Gravitational potential energy, PE = 2.8 J

Explanation:

It is given that,

Mass of the bluebird, m = 34 g = 0.034 kg

It flies from the ground to the top of an 8.5-m tree, h = 8.5 m

We need to find the change in the bluebird's gravitational potential energy as it flies to the top of the tree. It can be calculated as :

[tex]PE=m\times g\times h[/tex]

[tex]PE=0.034\ kg\times 9.8\ m/s^2\times 8.5\ m[/tex]

PE = 2.83 J

or

PE = 2.8 J

So, the gravitational potential energy as it flies to the top of the tree is 2.8 J. Hence, this is the required solution.

Answer:

NO

Explanation:

The answer is NO because when the particle reaches the center of square there is a net force acting on the particle dew to various other charges and this net force gives acceleration to the particle. Moreover, For particle or object to be in equilibrium the net force acting on it should be zero and hence no acceleration. Although velocity can be zero or non zero at equilibrium state.

Answer:

The context is missing here, but ill try to explain a general case.

Something is in equilibrium if it is in a valley of the potential energy, this is because things in life try to be in the minimal energy state possible. Think for example in a thing that is away from the ground, the object will try to reach the ground, in this way minimizing the potential energy.

Now, if once the particle reaches the center of the square it remains at rest, it means that the total forces acting on the particle are zero and this is why the particle stays at rest, this would mean that the particle is in equilibrium, and if someone moves it a little bit of the center, some of the forces will increase and others will decrease, and then the equilibrium will be broken and the particle will move again.

In another case, if the particle is momentarily at rest (just for a few seconds) it may be because the forces acting on it are affecting the particle in such way that is moving is fully stopped in one direction, and the new forces are accelerating the particle in the opposite direction (in the same way that if you throw something upside when it reaches the maximum height it has for a brief moment a velocity equal to zero)  

Answer:

[tex]T_2[/tex] = [tex]90.667K[/tex]

Explanation:

Given:

For the first solenoid

Number of turns, n₁ = 1200 turns/m

Current, I₁ = 2.5 A

Paramagnetic material temperature, T₁ = 320 K

Now for the second solenoid

Number of turns, n₂ = 1000 turns/m

Current, I₂ = 0.85 A

Paramagnetic material temperature = T₂

The magnetic flux (B) is given as

[tex]B=\frac{c\mu_onI}{T}[/tex]

where,

c = curie's constant

μ₀ = arithmetic constant

also it is given that the magnetization in both the cases are same

therefore the magnetic flux will also be equal

thus,

[tex]\frac{c\mu_on_1I_1}{T_1}[/tex] = [tex]\frac{c\mu_on_2I_2}{T_2}[/tex]

or

[tex]\frac{n_1I_1}{T_1}[/tex] = [tex]\frac{n_2I_2}{T_2}[/tex]

or

[tex]\frac{1200\times 2.5}{320}[/tex] = [tex]\frac{1000\times 0.85}{T_2}[/tex]

or

[tex]9.375[/tex] = [tex]\frac{850}{T_2}[/tex]

or

[tex]T_2[/tex] = [tex]\frac{850}{9.375}[/tex]

or

[tex]T_2[/tex] = [tex]90.667K[/tex]