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As shown below, a bullet of mass m and speed v is fired at an at rest pendulum bob. The bullet goes through the bob, and exits with a speed of v/4 . The pendulum bob is attached to a rigid pole of length L and negligible mass. What is the minimum value of v such that the pendulum bob will barely swing through a complete vertical circle? (Use the following as necessary: m, L, g, and M for the mass of the bob.)

As shown below a bullet of mass m and speed v is fired at an at rest pendulum bob The bullet goes through the bob and exits with a speed of v4 The pendulum bob class=

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leena

Hi there!

For an object to complete a circle, the minimum velocity can be derived by looking at the forces at the top of a circle:

∑F = Mg (tension goes to 0 N because this is the MINIMUM speed required)

Mv²/r = Mg

v² = gr

v = √gr

This is an example of a collision, so we can utilize the conservation of momentum:

Pi = Pf

P = mv

There is only one object moving initially, so:

Pi = mv

After the collision, the objects are moving as such:

Pf = M√gL + mv/4

Set the two equal:

mv = M√gL + mv/4

Multiply all terms by 4:

4mv = 4M√gL + mv

Solve for v:

3mv = 4M√gL

[tex]\large\boxed{v = \frac{4M\sqrt{gL} }{3m}}[/tex]

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