Classical Mechanics
# Simple Harmonic Motion

A body of mass \( m = 4 \text{ kg}\) is attached to a spring that hangs from the ceiling. If the spring is held at its original length and then let go, the body oscillates with amplitude \( A = {\frac{a}{b}} \text{ m}, \) where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b.\)

The spring constant is \( k = 5 \text{ N/m}\) and the gravitational acceleration is \(g=10\text{ m/s}^2.\)

A ring of mass \( M = 5 \text{ kg}\) and radius \( R = 3 \text{ m}\) oscillates on a horizontally placed rod. If the ring's angular frequency can be expressed as \( \omega = \sqrt{\frac{a}{b}} \text{ rad/s}, \) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b?\)

The moment of inertia is given by \( I = 2MR^2 \) and the gravitational acceleration is \(g=10\text{ m/s}^2.\)

A body of mass \( m = 8 \text{ kg}\) oscillates on a spring hanging from the ceiling with an angular frequency of \( \omega = \sqrt{\frac{a}{b}} \text{ rad/s}, \) where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b.\)

The spring constant is \( k = 5 \text{ N/m}\) and the gravitational acceleration is \(g=10\text{ m/s}^2.\)

In the figure above, a body of mass \( m = 4 \text{ kg}\) oscillates in the middle of two identical springs with an angular speed of \( \omega = \sqrt{\frac{a}{b}} \text{ rad/s}, \) where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b.\)

The spring constant is \( k = 9 \text{ N/m}\) and the gravitational acceleration is \(g=10\text{ m/s}^2.\)

A pendulum of mass \( m = 5 \text{ kg}\) and length \( L = 12 \text{ m}\) oscillates with an angular frequency of \( \omega = \sqrt{\frac{a}{b}} \text{ rad/s}, \) where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b.\)

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

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