1. The Yukawa Potential energy function for the interaction of neutrons and prot

Question

1. The Yukawa Potential energy function for the interaction of neutrons and protons in a nucleus is . (a) What is the force associated with this potential energy? (b) If and Find the value of the force at r=a and at r=3a

2. For what value of the constant c is the force conservative?

3. In a two-stage rocket, the exhaust velocity for the first stage burn is 3000 m/s and for the second stage burn is 3000 m/s. Assume that during the entire sequence of burns the rocket is still near enough to the surface of the earth to approximate the external gravitational acceleration as 9.8m/s^2 , and ignore air resistance. The mass of the first stage engine and tanks is 2000kg. The mass of the fuel held in the first stage is 4000kg. The first stage fuel burns in 10 seconds. The first stage is then ejected. The mass of the second stage engine, tanks, and payload is 2000kg. The mass of the fuel held in the second stage is 2000kg. The second stage fuel burns in 5 seconds. (The second stage tanks remain attached to the payload when empty.) (a) At what speed is the second stage/payload combination moving at the end of the second stage burn? (b) How much faster is this than the speed if all the 6000kg fuel were burned in the first stage in 15 seconds with the same exhaust velocity (3000m/s) and the entire 4000kg assembly of the empty rocket remained in one piece?

4. A particle with mass 10kg starts from rest at r = 10 and ?=0 radians, following the trajectory given by r=10-2t , ?=0.2t. Find the radial and angular components and of the force, as functions of time, that will cause this motion

5. A sheet of material, mass 2kg, uniform density, is in the shape of a right triangle bounded by the lines y=0, x=0, and y=4-x. It attched to the origin, and free to rotate about the y axis. A force of 1000 Newtons is applied perpendicular to the plane, located at the x=4 point on the triangle as it starts to rotate. In a right-handed coordinate system, how fast will the object be spinning (in radians per second) as it crosses the y-z plane (that is, when it has rotated through 270 degrees)?

Thank you in advanced.

Explanation / Answer

Force = - dU/dr to take the derivative, first pull     - aUo   out in front and you have    Force = a Uo  (d/dr) [ r-1 e-r/a ]    the derivative of the stuff in the brackes canjust be found using chain rule           -r-2 e-r/a   - (1/a)r-1 e-r/a   So     Force = a Uo [ - r-2 e-r/a   - (1/a)r-1 e-r/a   ]    or      Force =   - Uo   e-r/a  [   a  +  r ] /r2 at r=a =- 2Uoe-1/a at r=2a =- 12Uoe-2/a Force = - dU/dr to take the derivative, first pull     - aUo   out in front and you have    Force = a Uo  (d/dr) [ r-1 e-r/a ]    the derivative of the stuff in the brackes canjust be found using chain rule           -r-2 e-r/a   - (1/a)r-1 e-r/a   So     Force = a Uo [ - r-2 e-r/a   - (1/a)r-1 e-r/a   ]    or      Force =   - Uo   e-r/a  [   a  +  r ] /r2 at r=a =- 2Uoe-1/a at r=2a =- 12Uoe-2/a

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