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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?

Short Answer

Expert verified

The minimum initial speed of dart is 58.55 m/s

Step by step solution

01

Determination of speed of combination of ball and sphere to complete circular loopGiven Data:

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

The mass of the sphere is: M=20kg

The mass of the dart ball is: m=5kg

02

Concept

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=gl

Here, g is the gravitational acceleration.

u=9.8m/s22.8mu=5.24m/s

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

v=u2+4gl

Substitute all the values in the above equation.

v=5.24m/s2+49.8m/s22.8mv=11.71m/s

03

Determination of minimum initial speed of dart

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

m+Mv=mV

Substitute all the values in the above equation.

5kg+20kg11.71m/s=5kgVV=58.55m/s

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

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