Conservation of energy and centripital force

[ap.p.b.studen]

1. Homework Statement
Jin is sitting on top of a hemispherical, frictionless igloo of radius 2.40 meters. His friend pushes him, giving him an initial speed. Jin slides along the igloo and loses contact with it after he has traveled 1.60 meters along the surface. What was his initial speed?


2. Homework Equations
KEi+PEi=KEf+PEf

Ei=Ef

F=macos(angle)

Work=KE+PE

Work=Fx(delta x)

S=(angle)(radius)

PE=mgh

KE=.5(m)(v^2)

Fc=mv^2/r


3. The Attempt at a Solution

I drew a picture of a semicircle and used the S=theta(r) equationand trignometry to find that the hieghts (2.40m initial and 1.89m when leaves igloo). I used the fact that the centripal force would be equal to the force of gravity to find the velocity when jin left the igloo(4.85m/s) and used the conservation of energy to find that the inital velocity was 3.68m/s. This was wrong.?

 

[AvroArrow]

4. The Attempt at a Solution Use the conservation of energy equation and the fact that the force of gravity will equal the centripetal force to find the velocity when Jin leaves the igloo. PEi + KEi = PEf + KEFmghi + .5mv^2i = mghf + .5mv^2f2.40*9.8*m + .5m(vi)^2 = 1.89*9.8*m + .5m(vf)^2vi^2 = vf^2 + 2ghi - 2ghfvi = (vf^2 + 2ghi - 2ghf)^1/2vf=mv^2/r vf=(m)(vf^2)/2.4vi = [(m)(vf^2)/2.4)^2 + 2(9.8)(2.40) - 2(9.8)(1.89)]^1/2 vi = 3.68m/s

 

[kerimek]

I would like to clarify that conservation of energy and centripetal force are two separate concepts that are related to each other in this scenario. Conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. In this case, the initial kinetic energy of Jin is converted into potential energy as he slides up the igloo and then back into kinetic energy as he slides down.

On the other hand, centripetal force is the force that keeps an object moving in a circular path. In this scenario, the centripetal force is provided by the normal force of the igloo pushing against Jin as he slides along its surface. This force must be equal to the force of gravity acting on Jin to keep him in a circular path.

To solve this problem, you correctly used the conservation of energy equation to find Jin's initial velocity. However, your calculation for the centripetal force was incorrect. The correct equation for centripetal force is F = mv^2/r, where m is the mass of the object, v is the velocity, and r is the radius of the circular path. In this case, the radius would be the radius of the igloo, which is given as 2.40 meters. Using this equation, you can solve for the initial velocity of Jin.