When different instruments lie on a vibrating surface of the machine tool then at certain frequencies and amplitudes these instruments may jump to a stationary height. This looks like as if the instruments would "hang" over the surface. Give a theoretical explanation of the effect.

The small ball begins to roll without initial speed from a top of absolutely smooth hemisphere of radius R.  At what height it will come off a surface?

The answer: 2R/3

The cylinder of radius R lays on two thin rods. With what relative speed V the rods should move apart each other that falling of the cylinder would occur without a contact with them?

The answer: V > 2ÖgR

With what horizontal speed the ball should move on the top step of the stairs to hit the middle and bottom steps only one time as shown in animation. Width and height of the stairs is b.

The answer: (1/3Ö2 .. Ö2-1) Ögb

Splinter shell moves with the velocity u toward a flat wall. At the distance L from the wall the shall explodes and  falls to many pieces moving in all the directions with the velocity v relatively the centre of mass of the shall. Which area on the wall will be damaged by the fragments of the shall? Neglect the gravity and air resistance.

The answer: R = L (1 - (v/u)²)^1/2 / (u/v - v/u) when u > v and R = ∞ if u ≤ v

Ball of mass m moves with velocity u₀ toward the still rod of mass M = 2m and length 2L. Ball moves perpendicular to rod and collides it at the distance l from the center. After collision the velocity of the ball is u₁, rods moves with velocity V and rod rotates with angle frequency ω. Find 1) at which l the ball after collision will stop 2) velocity of the ball, rod and the angle frequency of the rod, if the ball strikes the rod at the end.

The answer: 1) l = L/√3   2) u₁ = V = u₀/3 and ω = u₀/L

Particles are emitted in all the directions from a point source in the field of gravitation and fall on a plate situated at the distance h below the source. Velocity of every particle is u₀.  1) What is an envelope of the region filled with the particles (in the plane XZ). 2) What is the volume V of the space filled with the particles if u₀²=2gh

The answer: z = u₀²/2g - gx²/2u₀²; V = pu₀⁶/g³

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