Mechanics dynamics

Solved examples from dynamics. Dynamics is a part of mechanics in which causes and results of motion are examined.

Material slides without friction from slope

Mechanics dynamics - example 1.

At the peak of the slope is material point. On time t=0 material point starts sliding down. Slope has triangle shape and stands on the ground. There is no friction between ground and slope’s base. Gravity field works on slope and material point. Gravity field is described by gravity acceleration g. Subject of example is to find road S which slope has traveled during material point sliding from it.

Forces between material point and slope during material point’s slide

Orb rolls from slope

Mechanics dynamics - example 2.

Orb rolls down from slope. Slope is inclined to the ground by angle α. Gravity field works on orb. Orb’s moment of inertia is known. There is sliding friction between orb and slope’s surface. During rolling orb spins around axis which is the middle of it. Spin axis is also parallel to slope’s surface.

Orb rolls down from slope

Rod during rotary motion

Mechanics dynamics - example 3.

Rod has length l and mass m. Rod stands in vertical position. In one moment rod’s end which is on top starts moving down around axis which is placed in rod’s second end. Rod is inside gravity field. Note that rod has mass m. It means that rod is rigid body so we can’t treat it like material point. Rod’s center of gravity is marked in its half of length. Subject of example is to find final velocity of rod’s end when it will be on the ground.

Dynamics of rod during rotary motion

Weights hanged through movable block

Mechanics dynamics - example 4.

Two weights are hanged via rope through movable block. Weights and block have mass. Rope is mass-less. Mass of one weight is bigger from second one. In moment weights start moving. One weight moves to up. Second weight moves down. Movement take place in gravity field. There is no friction between rope and block surface. We assume that air resist doesn’t exist. Because there is no friction on block’s surface no heat is released. During block’s motion rotate around axis in its middle. Block is a roller. Subject of example is to find acceleration of weight which moves down.

Acceleration of masses system

Mechanics dynamics – example 5

Mechanics dynamics - example 5.

Thread is fixed to the ceiling. Thread has a roller shape. Roller radius is r[m]. Mass m[kg] of thread is known. Thread is unrolling to down direction vertically. In example linear acceleration a[m/s2] has to be calculated. Thread is a roller with twine which was coiled on it. Mass of twine is so small so it is possible to assume it as equal to zero. Thread is a rigid body.

Mechanics dynamics – example 5

Mechanics dynamics – example 6

Mechanics dynamics - example 6.

Human is climbing on the rope in up direction. Earth gravity field works on human. Human has mass m[kg]. Rope is assumed as mass-less. Subject of example is designation of rope’s tension reaction force. In example the second principle of dynamics is applied.

Mechanics dynamics – example 6

Mechanics dynamics – example 7

Mechanics dynamics - example 7.

Roller is rolling to the slope’s top with start velocity v0. Roller executes flat motion because it is rotating around its axis of symmetry and moves progressive motion. Between roller contact surface with slope surface exists rolling friction. Rolling friction is described by rolling friction factor ƒ[m]. Roller has mass m[kg] and radius r[m]. Slope’s inclination angle α is known. Subject of example is to find road s which roller has traveled until it had stopped. The principle of conservation of total mechanical energy will be applied.

Mechanics dynamics – example 7

Mechanics dynamics – example 8

Mechanics dynamics - example 8.

Thin rod is fixed to the ceiling on one of its ends. End of rod is fixed to the ceiling by joint. Rod has length l[m] and mass m[kg]. Rod executes rotary motion around joint. Angular velocity ω is known. Rod is placed in Earth’s gravity field. Subject of example is designation of maximum force which works in rod’s center of mass. Rod is under influence of centrifugal force and gravity force.

Mechanics dynamics – example 8

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