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How far to the right of where the string was cut does the ball hit the floor?

A 110 g ball on a 60cm long string is swung in a vertical circle about a point 200 cm above the floor. The tension in the string when the ball is at the very bottom of the circle is 4.4 N. At the very bottom of the circle, a very sharp knife is suddenly inserted, to cut the string directly below the point of support

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The Best Answer for How far to the right of where the string was cut does the ball hit the floor?

Let:

T be the tension in the string,

m be the mass of the ball,

g be the acceleration due to gravity,

u be the horizontal speed of the ball when the string is cut,

r be the radius of the circle,

x be the horizontal distance to where the ball lands,

t be the time between cutting and landing,

h be the height of the ball above the floor when the string is cut.

T - mg = mu^2 / r ...(1)

x = ut ...(2)

h = gt^2 / 2 ...(3)

Eliminating t from (2) and (3):

x^2 = 2hu^2 / g

From (1):

u^2 = r(T / m - g)

Eliminating u^2:

x^2 = 2hr(T / m - g) / g

= 2hr(T / [ mg ] - 1)

x = sqrt{ 2 * (2.00 - 0.60)0.60(4.4 / [0.110 * 9.81] - 1) }

= 2.27 m.

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