#explain pls (A/S mech)

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spring edge
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Using F=ma, I understand how the first equation is formed

spring edge
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I understand how the second equation is formed.

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What did they do after?

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Not substitution.

sweet shoal
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there is no friction on particle p

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as it is a smooth surface

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there is a tension force (forward force) acting on p

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as you know, assumptions are; pulley = equal acceleration, light inextensible string = equal tension, smooth = no friction