Chapter 1: Q84P (page 269)

A 20.00-kg lead sphere is hanging from a hook by a thin wire 2.80 m long and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00-kg steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?

Expert verified

The minimum initial speed of dart is 58.55 m/s

Step by step solution

The length of wire for the block is $l = 2.8 m$

The mass of the sphere is: $M = 20 k g$

The mass of the dart ball is: $m = 5 k g$

The minimum initial speed of the dart is found by using the condition for a circular loop.

The maximum speed of the sphere is given as:

$u = \sqrt{g l}$

Here, g is the gravitational acceleration.

$u = \sqrt{(9.8 m / s^{2}) (2.8 m)} u = 5.24 m / s$

The speed of the combination of sphere and ball is given as:

$v = \sqrt{u^{2} + 4 g l}$

Substitute all the values in the above equation.

$v = \sqrt{{(5.24 m / s)}^{2} + 4 (9.8 m / s^{2}) (2.8 m)} v = 11.71 m / s$

Apply the conservation of momentum to find the minimum initial speed of dart.

$(m + M) v = m V$

Substitute all the values in the above equation.

$(5 k g + 20 k g) (11.71 m / s) = (5 k g) V V = 58.55 m / s$

Therefore, the minimum initial speed of dart is 58.55m/s

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