Ratios of Initial to Final Energy/Momentum/Velocity

In summary, a rotating star collapses under the influence of gravitational forces to form a pulsar with a radius of 4.00 x 10-4 times its original size and no change in mass. The star's mass is uniformly distributed in a spherical shape in both cases. The ratios of the angular momentum, angular velocity, and rotational kinetic energy after collapse to before collapse can be calculated. If the period of the star's rotation before collapse is 5.00 x 107 s, its period after collapse can be determined.
  • #1
Jtappan
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0

Homework Statement



A rotating star collapses under the influence of gravitational forces to form a pulsar. The radius of the star after collapse is 4.00 10-4 times the radius before collapse. There is no change in mass. In both cases the mass of the star is uniformly distributed in a spherical shape.
(a) What is the ratio of the angular momentum of the star after collapse to before collapse?


(b) What is the ratio of the angular velocity of the star after collapse to before collapse?


(c) What is the ratio of the rotational kinetic energy of the star after collapse to before collapse?


(d) If the period of the star's rotation before collapse is 5.00 107 s, what is its period after collapse?
____ s


Homework Equations





The Attempt at a Solution



I am having trouble imputing this into my online homework. Could you give me an example as to how I could put this into my online homework??
 
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  • #2
I have no idea how to put this into your online homework. You might start by filling in some blanks.
 
  • #3


I would approach this problem by first considering the conservation of angular momentum, energy, and velocity in a system. Since there is no change in mass, we can assume that the initial and final angular momentum, energy, and velocity of the star will be equal.

(a) The ratio of the angular momentum after collapse to before collapse can be calculated using the equation L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. Since the mass is uniformly distributed in a spherical shape, we can use the equation for the moment of inertia of a solid sphere, I = (2/5)mr^2. Therefore, the ratio of angular momentum is:

L_after/L_before = (I_after * ω_after)/(I_before * ω_before) = (2/5 * m * r_after^2 * ω_after)/(2/5 * m * r_before^2 * ω_before) = (r_after/r_before)^2 = (4.00*10^-4)^2 = 1.60*10^-7

(b) The ratio of angular velocity can be calculated using the equation ω = v/r, where v is the linear velocity and r is the radius. Since the mass and radius do not change, the linear velocity will also remain the same. Therefore, the ratio of angular velocity is:

ω_after/ω_before = (v_after/r_after)/(v_before/r_before) = (v_after/v_before) * (r_before/r_after) = (r_before/r_after) = 1/(4.00*10^-4) = 2.50*10^3

(c) The ratio of rotational kinetic energy can be calculated using the equation KE = (1/2)Iω^2. Using the same moment of inertia as before, the ratio of kinetic energy is:

KE_after/KE_before = [(1/2) * (2/5 * m * r_after^2) * ω_after^2]/[(1/2) * (2/5 * m * r_before^2) * ω_before^2] = (r_after/r_before)^2 * (ω_after/ω_before)^2 = (4.00*10^-4)^2 * (2.50*10^3)^2 = 4.00*10^-3

(d) Finally, to
 

1. What is the importance of studying ratios of initial to final energy/momentum/velocity?

Studying ratios of initial to final energy/momentum/velocity is important because it allows us to understand the transfer and conservation of energy and momentum in a system. This is crucial in various fields of science, such as physics and engineering, as it helps us predict and analyze the behavior of objects and systems.

2. How do you calculate the ratio of initial to final energy/momentum/velocity?

The ratio of initial to final energy/momentum/velocity can be calculated by dividing the final value by the initial value. For example, if the initial velocity of an object is 10 m/s and the final velocity is 20 m/s, the ratio would be 20/10 = 2.

3. What factors can affect the ratio of initial to final energy/momentum/velocity?

There are several factors that can affect the ratio of initial to final energy/momentum/velocity, such as external forces acting on the system, the mass and velocity of the objects involved, and the type of collision or interaction between the objects.

4. Can the ratio of initial to final energy/momentum/velocity ever be greater than 1?

In most cases, the ratio of initial to final energy/momentum/velocity will be less than 1, as energy and momentum are conserved in a closed system. However, in some cases, such as elastic collisions between objects, the ratio can be greater than 1 due to the transfer of kinetic energy from one object to another.

5. How does understanding ratios of initial to final energy/momentum/velocity contribute to scientific advancements?

Understanding ratios of initial to final energy/momentum/velocity is crucial in many scientific advancements, such as developing more efficient and safe transportation systems, designing better sports equipment, and analyzing the behavior of particles in particle accelerators. It also allows us to make accurate predictions and calculations in various fields of science and engineering.

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