Theoretical Mechanics DYNAMICS Technical University of Sofia Branch Plovdiv Theoretical Mechanics DYNAMICS * Navigation: Right (Down) arrow – next slide Left (Up) arrow – previous slide Esc – Exit Notes and Recommendations: ruschev@tu-plovdiv.bg

Lecture 3 Dynamics of a non-free particle Non-free particle A particle is called non-free (or constrained) if it can not occupy an arbitrary position in the space. Connections - the bodies, which restrict the motion of the particle and impose certain dependences on its displacements and velocities.

Lecture 3 Dynamics of a non-free particle Degrees of freedom First definition: the number of independent parameters that define the configuration of a mechanical system. Second definition: independent displacements and/or rotations that specify the orientation of the body or system. 3 degrees of freedom for a free particle

Lecture Dynamics of a non-free particle Principle of releasing The motion of the non-free particle does not change, if it is released from the imposed connections and it is considered as a free particle under the action of the active forces and the reaction of the removed connection. The vector equation of motion: where - tangential reaction or friction force - normal reaction

Lecture 3 Dynamics of a non-free particle Natural differential equations of a non-free particle along an ideal curve

Lecture 3 Sample Problem - 1 The m=60-kg skateboarder coasts down the circular track. If he starts from rest when q = 0°, determine the magnitude of the normal reaction the track (r=4 m) exerts on him when q = 60°. Neglect his size for the calculation. SOLUTION Free-Body Diagram G=m.g Equations of Motion.

Lecture 3 Sample Problem - 1 SOLUTION G=m.g

Lecture 3 Sample Problem - 2 Determine the maximum speed that the jeep can travel over the crest of the hill and not lose contact with the road. Determine the normal force the driver exerts on the seat of the car when the car traveling with a speed of 60 km/h if the driver mass is 85 kg.

Lecture 3 Sample Problem - 3 On a block M of mass m is given an initial velocity v0, when it is at the top of a smooth sphere. The sphere radius is r. Determine the angle jm and the velocity vm at the place, where it will leave the sphere’s surface.