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Honey Bunison

Honey B.

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Estimate the ideal gas pressure and the radiation pressure at the center of Sirius $\mathrm{B}$, using $3 \times$ $10^{7} \mathrm{K}$ for the central temperature. Compare these values with the estimated central pressure, Eq. (1). $$P_{c} \approx \frac{2}{3} \pi G \rho^{2} R_{\mathrm{wd}}^{2} \approx 3.8 \times 10^{22} \mathrm{N} \mathrm{m}^{-2}$$

An Introduction to Modern Astrophysics

The neutrino flux from SN 1987 A was estimated to be $1.3 \times 10^{14} \mathrm{m}^{-2}$ at the location of Earth. If the average energy per neutrino was approximately $4.2 \mathrm{MeV}$, estimate the amount of energy released via neutrinos during the supernova explosion.

The neutrino flux from SN 1987 A was estimated to be $1.3 \times 10^{14} \mathrm{m}^{-2}$ at the location of Earth. If the average energy per neutrino was approximately $4.2 \mathrm{MeV}$, estimate the amount of energy released via neutrinos during the supernova explosion.

An Introduction to Modern Astrophysics

During the Great Eruption of $\eta$ Car, the apparent visual magnitude reached a characteristic value of $m_{V} \sim 0 .$ Assume that the interstellar extinction to $\eta$ Car is 1.7 magnitudes and that the bolometric correction is essentially zero. (a) Estimate the luminosity of $\eta$ Car during the Great Eruption. (b) Determine the total amount of photon energy liberated during the twenty years of the Great Eruption. (c) If $3 \mathrm{M}_{\odot}$ of material was ejected at a speed of $650 \mathrm{km} \mathrm{s}^{-1}$, how much energy went into the kinetic energy of the ejecta?

An Introduction to Modern Astrophysics

Estimate the neutron degeneracy pressure at the center of a $1.4 \mathrm{M}_{\odot}$ neutron star (take the central density to be $1.5 \times 10^{18} \mathrm{kg} \mathrm{m}^{-3}$ ), and compare this with the estimated pressure at the center of Sirius B.

An Introduction to Modern Astrophysics

Questions asked

INSTANT ANSWER

a 1d potential energy function is defined as V(x)= (1-(x^2/b^2))^(-1/2). a) find the force function F(x) for this potential. b) an object possessing kinetic energy K is released from the origin in this potential. Find its maximum displacement from the origin in terms of K and b

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AWAITING AN EDUCATOR

2. Potential Well A \( 1 \mathrm{D} \) potential energy function is defined as \( V(x)=\left(1-\frac{x^{2}}{b^{2}}\right)^{-1 / 2} \). a) Sketch \( V(x) \) in the range \( -b<x<b \). b) Find the force function \( F(x) \) for this potential. c) An object possessing kinetic energy \( K \) is released from the origin in this potential. Find its maximum displacement from the origin in terms of \( K \) and \( b \). d) For an object of mass \( m \), find the angular frequency of oscillations for small displacements around the minimum of \( V(x) \).

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INSTANT ANSWER

1. Rocket Car A rocket car releases combustion gases at a constant speed of \( v_{0} \) with respect to itself and the mass of the gas released per unit time is \( \mu \). Assume that the car is initially at rest. The initial mass of the car with the combustion fuel is \( M_{0} \). a) Write down an expression for the total mass of the car including combustion fuel, \( m(t) \), as a function of time \( t \). b) Given that momentum of the gas \( \left(p_{\text {gas }}\right) \) is the product of the mass of the gas and its speed, calculate the momentum per unit time \( \left(d p_{g a s} / d t\right) \) of the combustion gas released. c) Given that the momentum per unit time is the same as the net force (i.e. \( F_{\text {net }}=d p_{g a s} / d t \) ), find the velocity of the car as a function of time \( (v(t)) \).

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INSTANT ANSWER

An object of mass m is released with an initial velocity v = v_0 and experiences gravitational force and a linear drag force as it falls, leading to a terminal speed. The net force acting on the object is given by F_net = -mg - bv where b is a positive constant and g - 9.8 m/s^2 is the acceleration of free-fall. Assume that the downward direction is negative x-axis. a) Find the velocity as a function of time v(t).

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ANSWERED

Supreeta N verified

Numerade educator

A rope of length 𝐿 and uniform linear density 𝜌 is hanging over the edge of a table such that the vertically hanging piece has a length π‘₯0 at time 𝑑 = 0. The position of the bottom end of the rope, i.e. point 𝑃, with respect to the edge of the table is given at time 𝑑 by π‘₯(𝑑). Initially (at 𝑑 = 0), the rope is given an upward velocity 𝑣(0) = βˆ’π‘£0, where 𝑣0 is a positive constant. Assume that point 𝑃 does not reach the ground at any time in the problem. a) Determine the net force (𝐹𝑛𝑒𝑑) acting on the rope in terms of 𝜌, π‘₯, 𝑔 (ignore friction and the resistance of the rope to bending). b) Find the speed of the rope 𝑣(π‘₯) as a function of π‘₯ in terms of 𝐿, 𝑔, π‘₯0. c) Find π‘₯(𝑑) in terms of π‘₯0, 𝑔, 𝐿 (hint: replace 𝐹𝑛𝑒𝑑 in part (a) with π‘šπ‘₯̈ ). Use the initial conditions provided to find the unknown constants. d) Find the time when π‘₯(𝑑) = 0. e) Determine the minimum initial speed 𝑣0 that allows the entire chain to climb on top of the table. f) Show that if 𝑣0 is 𝛼 times larger than the speed found in (e), then the time required for the entire chain to climb on to the table is 𝑑 = 1/2 √𝐿/𝑔 ln (1+𝛼/π›Όβˆ’1)

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INSTANT ANSWER

An airplane of mass π‘š is flying in a horizontal circle at an altitude β„Ž above the ground. The point on the ground located directly below the center of the circle is considered the origin of a 3D coordinate system, where the +𝑧-axis is the line passing through the origin to the center of the circle. The angle between the 𝑧-axis and the dotted line in the figure is 𝛼.The airplane experiences a lift force (𝐹𝐿⃗⃗⃗⃗ ) from the air that acts in a direction perpendicular to the dotted line and a downward gravitational force (π‘Šβƒ—βƒ—βƒ— ). a) Find the radius π‘Ÿ of the horizontal circular path of the object in terms of β„Ž and 𝛼. b) Draw the two forces acting on the airplane in the diagram below: c) Using cylindrical coordinates, find the net force 𝐹𝑛𝑒𝑑⃗on the object (with components along π‘’π‘ŸΜ‚ , π‘’πœ™Μ‚ , 𝑒𝑧̂ ). d)By comparing 𝐹𝑛𝑒𝑑⃗⃗⃗⃗⃗⃗⃗⃗ with the acceleration vector in cylindrical coordinates, determine 𝐹𝐿 (in terms of π‘š, 𝑔, 𝛼) and determine the angular velocity πœ™Μ‡ (in terms of 𝑔, β„Ž, 𝛼). The airplane now travels in a path that spirals in towards the center of the circle while maintaining the same altitude β„Ž. The radius of the path changes at a constant rate of π‘ŸΜ‡ = 𝛽. e) Find πœ™Μ‡ in terms of β„Ž and π‘Ÿ. f) Find πœ™Μˆ in terms of β„Ž, π‘Ÿ, and 𝛽. g) Find the velocity vector in cylindrical coordinates. h) Write down the acceleration vector in cylindrical coordinates. i) Find the rate of change of the angle 𝛼 with respect to time, i.e. 𝛼̇ , in terms of β„Ž, π‘Ÿ, and 𝛽. j) Find 𝐹𝐿̇ in terms of β„Ž, π‘Ÿ, and 𝛽.

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INSTANT ANSWER

An airplane of mass \( m \) is flying in a horizontal circle at an altitude \( h \) above the ground. The point on the ground located directly below the center of the circle is considered the origin of a 3D coordinate system, where the \( +z \)-axis is the line passing through the origin to the center of the circle. The angle between the \( z \)-axis and the dotted line in the figure is \( \alpha \). The airplane experiences a lift force \( \left(\overrightarrow{F_{L}}\right) \) from the air that acts in a direction perpendicular to the dotted line and a downward gravitational force \( (\vec{W}) \). a) Find the radius \( r \) of the horizontal circular path of the object in terms of \( h \) and \( \alpha \). b) Draw the two forces acting on the airplane in the diagram below: c) Using cylindrical coordinates, find the net force \( \overrightarrow{F_{n e t}} \) on the object (with components along \( \widehat{e_{r}}, \widehat{e_{\phi}}, \widehat{e_{z}} \) ).

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INSTANT ANSWER

A body of mass moves in under the influence of a conservative force (i.e. a force that conserves potential energy in a closed loop trajectory) with potential energy where is a positive constant. a) Determine the angular frequency of oscillations for small displacements around this minimum in terms of the constants provided in this problem.

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INSTANT ANSWER

A body of mass \( m \) moves in \( 1 D \) under the influence of a conservative force (i.e. a force that conserves potential energy in a closed loop trajectory) with potential energy \( V(x)=-e^{-k x} \cos x \), where \( k \) is a positive constant. b) Find the coordinates of the first minimum of \( V(x) \) located in the region \( x>0 \). c) Sketch \( V(x) \) for \( x>0 \), making sure to indicate the minimum and the \( y \)-intercept. d) Find the Taylor expansion of \( V(x) \) around its minimum up to and including the term in \( x^{2} \). e) Determine the angular frequency of oscillations for small displacements around this minimum in terms of the constants provided in this problem. f) Suppose the particle has a speed of \( v_{0} \) at the minimum found in (b). Find the velocity \( v(x) \) as a function of position of the body.

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