Constrained Motion of Connected ParticlesHere 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.

𝑠=𝐿=𝑥+14

(2𝜋𝑟2 )+2 𝑦+12

(2𝜋𝑟 1 )+𝑏

To determine how the variable quantities change we will differentiate with respect to time.

𝑑𝑑𝑡 [𝐿=𝑥+ 1

4(2𝜋 𝑟2 )+2 𝑦+ 1

2(2𝜋𝑟 1)+𝑏  ]

→0=�̇�+2 �̇� Relates the velocities of the two particles.

Similarly,𝑑𝑑𝑡

[0=�̇�+2 �̇� ] →0=�̈�+2 �̈�

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.

𝐿𝐴=𝑦𝐴+2 𝑦𝐷+𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝐿𝐵=𝑦𝐵+ 𝑦𝑐+( 𝑦𝑐−𝑦 𝑑)+𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑑𝑑𝑡 [𝐿𝐴=𝑦𝐴+2 𝑦𝐷+𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ]

𝑑𝑑𝑡 [𝐿𝐵=𝑦𝐵+𝑦 𝑐+ (𝑦 𝑐− 𝑦𝑑 )+𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡   ]

→0=�̇� 𝐴+2 �̇�𝐷

→0=�̇�𝐵+2 �̇�𝑐− �̇�𝐷

→ �̇�𝐷=−12�̇�𝐴

Combining to get a single expression containing the velocities of the three particles of interest:

→0=�̇�𝐵+2 �̇�𝑐+12�̇�𝐴 →0=2 �̇�𝐵+4 �̇� 𝑐+ �̇�𝐴

Similarly for acceleration,𝑑𝑑𝑡 [0= �̇�𝐴+2 �̇�𝐷 ]

𝑑𝑑𝑡 [0= �̇�𝐵+2 �̇�𝑐− �̇�𝐷 ]

→0=�̈� 𝐴+2 �̈�𝐷

→0=�̈�𝐵+2 �̈�𝐶− �̈�𝐷

→0=2 �̈�𝐵+4 �̈�𝐶+ �̈�𝐴

Chapter 3Kinetics of Particles

Kinetics is the study of unbalanced forces and the resulting changes in motion. There are three primary analysis techniques:

A. Newton’s Second Law B. Work and EnergyC. Impulse and Momentum

Force, Mass and Acceleration:

Newton’s Second Law

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!