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## Homework Statement

A block with a mass M is located on a frictionless, horizontal surface and is attached to a horizontal spring with spring stiffness k. The block is being pulled out to the right a distance [itex]x=x_0[/itex] of equilibrium and released at [itex]t = 0[/itex].

At time [itex]t_1[/itex], corresponding to [itex]\omega t_1=\varphi_1[/itex], a lump of clay with mass m is dropped onto the block (sticking to it).

a) Use conservation of momentum, in the horizontal direction, to show that the new amplitude is:

[itex]x_0\prime = x_0 \sqrt{\frac{M+m\cdot cos^2(\varphi_1)}{M+m}}[/itex]

b) Express the new angular frequency, [itex]\omega\prime[/itex] in terms of [itex]\omega[/itex]

## Homework Equations

[itex]x(t)=x_0 \cdot cos(\omega\cdot t)[/itex]

[itex]E=\frac{1}{2}kx_0^2[/itex]

[itex]\omega=\sqrt{\frac{k}{m}}[/itex]

## The Attempt at a Solution

a) Conservation of momentum:

[itex]M\dot{x}=(M+m)\dot{x\prime}[/itex]

[itex]M(-\omega\cdot sin(\omega\cdot t)\cdot x_0) =(M+m)\dot{x\prime}[/itex], I'm not sure whether it should be [itex]sin(\omega\cdot t_1)[/itex] or [itex]sin(\omega\cdot t)[/itex]

[itex]-M\sqrt{\frac{k}{M}}\cdot sin(\omega\cdot t)\cdot x_0=(M+m)\dot{x\prime}[/itex]

[itex]-\sqrt{M\cdot k}\cdot sin(\omega\cdot t)\cdot x_0=(M+m)\dot{x\prime}[/itex], now I don't know what more to do. If I express [itex]\dot{x\prime}[/itex] using [itex]sin(\omega \cdot t)[/itex], ω and t will be different so i can't remove them later (unless i know the relationship between the angular frequencies, which is just what part b) is)

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