A rigid, uniform, horizontal bar of mass m_1 and length L is supported by two identical massless strings. Both strings are vertical. String A is attached at a distance d < L/2 from the left end of the bar and is connected to the ceiling; string B is attached to the left end of the bar and is connected to the floor. A small block of mass m_2 is supported against gravity by the bar at a distance x from the left end of the bar, as shown in the figure.
A). Find T_A, the tension in string A.
Express the tension in string A in terms of g, m_1, L, d, m_2, and x.
B). Find T_B, the magnitude of the tension in string B.
Express the magnitude of the tension in string B in terms of T_A, m_1, m_2, and g.
C). If the mass of the block is too large and the block is too close to the left end of the bar (near string B) then the horizontal bar may become unstable (i.e., the bar may no longer remain horizontal).
What is the smallest possible value of x such that the bar remains stable (call it x_critical)?
Express your answer for x_critical in terms of m_1, m_2, d, and L.
The solution shows how to calculate the tensions in the vertical strings, which are used to hang a vertical bar.
difference between the tensions in the string
A ball of mass m is attached to a string of length L. It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion. Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical circle, the ball's speeds are v_t and v_b, and the corresponding tensions in the string are T_t and T_b. T_t and T_b (vectors) have magnitudes T_t and T_b.
Find T_b - T_t, the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle.
Express the difference in tension in terms of m and g. The quantities v_t and v_b should not appear in your final answer.