Classical Mechanics
# Conservation of Energy

A ball of mass \( m = 5 \text{ kg } \) slides down from the top of a hemisphere of radius \( R = 6 \text{ m }.\) The speed of the ball when it loses contact with the hemisphere can be expressed as \( v = \sqrt{\frac{a}{b}} \text{ m/s}, \) where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b.\)

The gravitational acceleration is \( g= 10 \text{ m/s}^2. \)

A ball of mass \( m = 7 \text{ kg} \) falls towards the ground from a height of \( H = 8000 \text{ km} .\) When the ball's height is \( h = 3000 \text{ km} ,\) its speed can be expressed as \( v = \sqrt{\frac{GM\times a}{b}} \text{ km/s}, \) where \(a\) and \(b\) are coprime positive integers. If the air resistance is negligible, what is the value of \(a+b?\)

\(G\) is gravitational constant and \(M\) is the mass of the earth. The radius of the earth is \( R = 6000 \text{ km} .\)

A ball of mass \( m = 5 \text{ kg } \) slides down from the top of a frictionless hemisphere of radius \( R = 12 \text{ m} .\) At what height \(h\) does the ball lose contact with the hemisphere?

The gravitational acceleration is \( g= 10 \text{ m/s}^2. \)

A ball of mass \( m = 5 \text{ kg} \) falls down towards the ground from a height of \( h = 3000 \text{ km}. \) The speed with which the ball strikes the ground is \( v = \sqrt{\frac{GM\times{a}}{b}} \text{ km/s}, \) where \(a\) and \(b\) are coprime positive integers. If the air resistance is negligible, what is the value of \(a+b?\)

\(G\) is the gravitational constant and \(M\) is the mass of the earth. The radius of the earth is \( R = 6000 \text{ km} .\)

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