Ftotal=Fstatic+Fdynamiccap F sub t o t a l end-sub equals cap F sub s t a t i c end-sub plus cap F sub d y n a m i c end-sub Step 2: Calculate the Static Weight The linear mass density of the chain is .When the top of the chain has fallen a distance , a length of the chain is coiled on the table.
The force exerted on the table is exactly three times the weight of the chain resting on the table:
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vx,cm=−L2sinθ⋅θ̇v sub x comma c m end-sub equals negative the fraction with numerator cap L and denominator 2 end-fraction sine theta center dot theta dot Ftotal=Fstatic+Fdynamiccap F sub t o t a l
Since no external forces act on the system,
Using the Euler-Lagrange equation
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such that the ball immediately exhibits pure rolling without slipping from the instant of impact. Step 1: Set up Impulse Relations
a=−fkM=−μga equals the fraction with numerator negative f sub k and denominator cap M end-fraction equals negative mu g
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1/2mv2=mgdsinα+f⋅d1 / 2 m v squared equals m g d sine alpha plus f center dot d The potential energy at the top ( ) is converted back to kinetic energy ( ) and work done against friction (
be the linear impulse imparted by the cue stick to the ball.According to the linear impulse-momentum equation: J=Mv0cap J equals cap M v sub 0