Constrained Motion of Connected Particles Here we will explore the effects of constraint on the motion of connected objects. One Degree of Freedom: Degree of Freedom - The number of degrees of freedom corresponds to the number of variables required to specify completely the motion of a particle. In this case, the total possible distance that can be moved by either A or B is the length of the rope. Notice that there are some portions of the rope that contain fixed lengths, which limit the distance the particles can travel. We will therefore define the length of the rope in terms of the variable length and fixed length segments. 𝑑 𝑑𝑡 𝐿=𝑥+ 1 4 2𝜋 𝑟 2 +2𝑦+ 1 2 2𝜋 𝑟 1 +𝑏 Relates the velocities of the two particles. 𝑠=𝐿=𝑥+ 1 4 2𝜋 𝑟 2 +2𝑦+ 1 2 2𝜋 𝑟 1 +𝑏 →0= 𝑥 +2 𝑦 Similarly, To determine how the variable quantities change we will differentiate with respect to time. 𝑑 𝑑𝑡 0= 𝑥 +2 𝑦 →0= 𝑥 +2 𝑦 Relates the accelerations of the two particles.

Two Degrees of Freedom: Here we have two independent ropes, both of which are required to define the motion of the particles. 𝐿 𝐴 = 𝑦 𝐴 +2 𝑦 𝐷 +𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐿 𝐵 = 𝑦 𝐵 + 𝑦 𝑐 + 𝑦 𝑐 − 𝑦 𝑑 +𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑑 𝑑𝑡 𝐿 𝐴 = 𝑦 𝐴 +2 𝑦 𝐷 +𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 →0= 𝑦 𝐴 +2 𝑦 𝐷 → 𝑦 𝐷 =− 1 2 𝑦 𝐴 𝑑 𝑑𝑡 𝐿 𝐵 = 𝑦 𝐵 + 𝑦 𝑐 + 𝑦 𝑐 − 𝑦 𝑑 +𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 →0= 𝑦 𝐵 +2 𝑦 𝑐 − 𝑦 𝐷 Combining to get a single expression containing the velocities of the three particles of interest: →0= 𝑦 𝐵 +2 𝑦 𝑐 + 1 2 𝑦 𝐴 →0=2 𝑦 𝐵 +4 𝑦 𝑐 + 𝑦 𝐴 Similarly for acceleration, 𝑑 𝑑𝑡 0= 𝑦 𝐴 +2 𝑦 𝐷 →0= 𝑦 𝐴 +2 𝑦 𝐷 →0=2 𝑦 𝐵 +4 𝑦 𝐶 + 𝑦 𝐴 𝑑 𝑑𝑡 0= 𝑦 𝐵 +2 𝑦 𝑐 − 𝑦 𝐷 →0= 𝑦 𝐵 +2 𝑦 𝐶 − 𝑦 𝐷

Chapter 3 Kinetics of Particles

Kinetics is the study of unbalanced forces and the resulting changes in motion. There are three primary analysis techniques: Newton’s Second Law → 𝐹 =𝑚 𝑎 Work and Energy Impulse and Momentum Force, Mass and Acceleration: Newton’s Second Law → 𝐹 =𝑚 𝑎 This relationship is only valid for an inertial reference frame! Inertial reference frame – Non-Accelerating reference frame Newton’s second law is a second order differential equation. The dependence of F on time, position or velocity must be considered in the solution. Use the relationships we have developed for kinematics. You must consider all forces acting on the system, applied and reactive!