A uniform rod of mass $m_r = 168$ g and length $L = 100.0$ cm is attached to the wall with a pin as shown. Cords are attached to the rod at the $r_1 = 10.0$ cm and $r_2 = 90.0$ mark, passed over pulleys, and masses of $m_1 = 241$ g and $m_2 = 127$ g are attached. The Spice Girls have the following questions to ask you:
(a) Ginger Spice and Posh Spice want you to find the position $r_3$ on the rod where you would suspend a mass $m_3 = 200$ g in order to balance the rod and keep it horizontal if released from a horizontal position. In addition, for this case, what force (magnitude and direction) does the pin exert on the rod? Use standard angle notation to determine the direction of the force the pin exerts on the rod. Express the direction of the force the pin exerts on the rod as the angle $\theta_F$, measured with respect to the positive x-axis (counterclockwise is positive and clockwise is negative).
$r_3 = \text{____ m}$
$F_{pivot} = \text{____ N}$
$\theta_F = \text{____ }^\circ$
(b) Scary Spice and Sporty Spice tell you to remove the mass $m_3$ and determine the new mass $m_4$ that you would need to suspend from the rod at the position $r_4 = 20.0$ cm in order to balance the rod and keep it horizontal if released from a horizontal position. In addition, for this case, what force (magnitude and direction) does the pin exert on the rod? Express the direction of the force the pin exerts on the rod as the angle $\theta_F$, measured with respect to the positive x-axis (counterclockwise is positive and clockwise is negative).
$m_4 = \text{____ kg}$
$F_{pivot} = \text{____ N}$
$\theta_F = \text{____ }^\circ$
(c) Baby Spice now tells you to remove the mass $m_4$ and determine the mass $m_5$ that you would need suspend from the rod in order to have a situation such that the pin does not exert a pivot force on the rod at all and the location $r_5$ from which you would suspend this mass in order to balance the rod and keep it horizontal from a horizontal position.
$m_5 = \text{____ kg}$
$r_5 = \text{____ m}$