Bullet and Block Circular Problem

[matt-83]

Homework Statement

I am going through past Olympiad qualifying exams as practice for this year's test. I came upon the following problem in the 2008 F=ma and continue to come up with the wrong answer. I keep coming up with C and that is not the answer provided in the answer key.


22. A bullet of mass m1 strikes a pendulum of mass m2 suspended from a pivot by a string of length
L with a horizontal velocity v0. The collision is perfectly inelastic and the bullet sticks to the
bob. Find the minimum velocity v0 such that the bob (with the bullet inside) completes a circular
vertical loop.
(a) 2√Lg
(b) √5Lg
(c) (m1 + m2)2√Lg/m1
(d) (m1 − m2)√Lg/m2
(e) (m1 + m2)√5Lg/m1

Homework Equations

m1v1= (m1+m2)v2
1/2 mv ^2 = mgh

The Attempt at a Solution

m1v1 = (m1+m2)v2
v2 = (m1v1)/(m1+m2)

So the kinetic energy of the block should be:
1/2 (m1+m2)v2^2
Then I set the initial kinetic energy of the block equal to the potential energy it would have at height 2L.

1/2 (m1+m2)v2^2 = 2L(m1+m2)g

Since v2 = (m1v1)/(m1+m2):

1/2(m1+m2)*((m1v1)/(m1+m2))^2 = 2L(m1+m2)g

m1+m2 should cancel:

((m1v1)/(m1+m2))^2 = 4Lg

(m1v1)^2 / (m1+m2) ^2 = 4Lg

m1 ^2 * v1 ^ 2 = 4Lg / ((m1+m2) ^2)

Take the Square Root of the whole thing:

m1 * v1 = 2sqrt(Lg)/(m1+m2)

SO my answer is (c) (m1 + m2)2√Lg/m1, which is apparently incorrect.

If someone could help me correct this that would be great.

Thanks.

 

[rl.bhat]

Then I set the initial kinetic energy of the block equal to the potential energy it would have at height 2L.
At the hight 2L the block has kinetic energy as well as the potential velocity.
To find the velocity at the top for minimum tension( = 0), equate the centripetal force to the weight of the block.

 

[thomate1]

Dear student,

Thank you for sharing your work and thought process for this problem. It seems like you have approached the problem correctly and have used the appropriate equations. However, there may be a small error in your calculations. Let's take a closer look at your final equation:

m1 * v1 = 2sqrt(Lg)/(m1+m2)

The left side of the equation is correct, but the right side seems to be missing a square root. It should be:

m1 * v1 = 2sqrt(Lg)/√(m1+m2)

Therefore, your final answer should be (c) (m1 + m2)√Lg/√(m1+m2), which matches with the correct answer given in the answer key.

Keep up the good work and good luck on your upcoming test!