A)
Let T1 be the tension force in the upper string and T2 be the tension force in the lower string.
We can set up two separate equations using Newton's second law, with the forces acting in the x- and y-directions:
x-direction:
T2 - T1*sin(theta) = m*w^2*R
y-direction:
T1*cos(theta) - m*g = 0
Now we can solve for T1 and T2. From the y-direction equation, we have:
T1 = m*g / cos(theta)
Plugging this into the x-direction equation:
T2 - (m*g / cos(theta))*sin(theta) = m*w^2*R
Rearranging to solve for T2:
T2 = m*w^2*R + (m*g / cos(theta))*sin(theta)
Now we have the forces in terms of the given variables:
T1 = m*g / cos(theta)
T2 = m*w^2*R + (m*g / cos(theta))*sin(theta)
B)
For the lower string to barely remain taut, the tension force T2 should be close to zero.
T2 = m*w^2*R + (m*g / cos(theta))*sin(theta) ≈ 0
Rearranging to solve for w:
w^2 = (- g * sin(theta) / R) * cos(theta)
Since w must be real and positive, we have:
w(min) = sqrt((- g * sin(theta) / R) * cos(theta))
a ball of mass m is attached by two strings to a vertical rod (lower string is at a right angle to rod). the entire system rotates at a constant angular velocity w about the axis of the rod.
A) assuming w is large enough to keep both strings taut, find the force each string exerts on the ball in terms of w, m, g, R, and theta.
B) find the minimum angular velocity, w(min) for which the lower string barely remains taut
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