1 answer

A ball is tied to one end of a string. (Projectile Motion)

Question:

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A ball is tied to one end of a string. The other end of the string is fixed. The ball isset in motion around a vertical circle without friction. At the top of thecircle, theball has a speed of √gR, as shown in the figure below. At what angle θ should the string be cut so that the ball will travel through the center ofthe circle?

PS. The answer is 100.6 I need to show work.

Energy = 1/2 mgR + mg(2R)

B/C Energy is conserved

Energy = mgR(5/2) J

Please Help ME!!!


Answers

at given angle suppose it has tangential velocity v

then horizontal component = v*cos(180-θ) = -vcos(θ)

vertical component = vsin(180-θ) = vsin(θ)

Rcos(θ-90) = -vcos(θ)*T --> Rsin(θ)=-vcos(θ)*T

Rsin(θ-90) = vsin(θ)T+.5gT^2 ---> -Rcos(θ)=vsin(θ)T+.5gT^2

put T from the first equation into second

top it has KE = .5mRg
if it is at angle as shown then it is [R-Rsin(θ-90)] below the top point = R(1+sin(θ))

loss in PE =mgR(1+sin(θ))

KE at this point = .5mRg + mgR(1+sin(θ))

we will get v from this equation

equating v from both and solving will give the answer

I hope this may help you some what

.

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