Potential Energy (10 points)?

So, we know that the rock has to get up to the top, which is 6.4m *2 = 12.8m above the height at the bottom. Since it requires energy to move something up in a downward gravitational field, we'll need energy to do that. But, there's more. In order for the string to not go slack, the rock still needs enough speed at the top of the circle to "pull" away from the center of the circle with enough force to counter that of gravity. This means it needs kinetic energy too!

So, the necessary speed comes from v^2/r = g, where r is the length of the string (radius of circle), and g is gravity, v is velocity at the top. Call it vf, since it's the final velocity you'll need.

Conservation of energy tells us that:

KEi + PEi = KEf + PEf

In our case, PEi is zero, KEi comes from the initial velocity, KEf comes from the velocity that you need (calculated from the centripetal acceleration as shown earlier), and PEf is the final potential energy, obtained from moving up 12.8m in the gravitational field.

Hope it helps!

at the top Fc = Fg, mv^2/r = mg, v = sqrt(gr) = 7.9 m/s is the minimum velocity needed at the top of the circle, for a minimum of 7.9 m/s at the top, we have a velocity at bottom of:

v^2 = (v^2 at top) + 4g*6.4 = 17.7^2 m/s

so, vi >= 17.7 m/s, anything less, and it won't make it to the top.

We measure potential energy, PE, relative to the lowest position on the vertical circle. So there, the total energy TE = KE + PE = KE because PE = 0 when at the bottom of the circle.

So TE = KE = 1/2 mU^2 where U = vi the initial speed.

At the top of the circle, from the COE law, TE = ke + pe = 1/2 mv^2 + mgH = 1/2 mv^2 + mg 2R where R = 6.4 m radius. H = 2R is the height above the bottom of the circle. As the two total energies are the same, we can write....

1/2 mU^2 = 1/2 mv^2 + mg 2R; so that U^2 = v^2 + 4gR and radial acceleration v^2/R = g and v^2 = gR is a necessary condition for the cord to remain taut. That is, the radial acceleration offsets the gravity acceleration while atop the circle.

In which case, U^2 = v^2 + 4v^2 = 5v^2 so that v = U/sqrt(5) mps. ANS. You can do the math as you did not share U = vi with us.

NOTE the physics. The radius R does not figure into the atop speed.

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