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fresh_42
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Summary:

This is an experiment if we could establish a little physics competition.
Classical Physics, Special Relativity Theory
Main Question or Discussion Point
Problem 1 (@wrobel )
(solved by @TSny )
There is a perfectly rough horizontal table. This table is pretty wide (actually it is a plane) and it rotates about some vertical axis. Angular velocity is a given constant: ##\Omega\ne 0##. Somebody throws a homogeneous ball on the table. The ball has a mass ##m## and a radius ##r##.
The ball begins to roll on the table without slipping. Initially (##t=0##) a vector ##\boldsymbol{OP}## (see the picture) is known; initial angular velocity of the ball and initial velocity of its center of mass ##S## are also known vectors.
1) Prove that the center of the ball describes a circle relative to the lab frame;
2) Find the radius of this circle.
Problem 2 (@wrobel )
(solved by @TSny , @Chestermiller )
There are four homogeneous rods. Each rod has mass ##m## and length ##b##. The ends of the rods are connected by frictionless hinges such that the rods form a rhombic frame (see the picture). This frame is shaped as square ##ABCD## and put at rest on a smooth horizontal table. Then one applies a force ##F## to the hinge ##A## along the diagonal #AC#. Find acceleration of the point ##C## right after the force has been applied.
Problem 3 (@PeroK )
(solved by @Gaussian97 )
a) Suppose a particle has threevelocity ##\vec{u} = (u_x, u_y, u_z)## in frame ##S##, which is moving with velocity ##(v, 0, 0)## relative to frame ##S'##. Show that the gamma factor of the particle in frame ##S'## is given by:
$$\gamma'_u = \gamma_v \gamma_u(1 + \frac{vu_x}{c^2})$$
b) Imagine a rig, consisting of two vertical poles, a distance ##L## apart. Two synchronised clocks are mounted on the poles and moving upwards at speed ##u##, relative to the rig, remaining horizontally aligned with respect to the rig.
Imagine a frame of reference in which the whole rig is moving horizontally at speed ##v##. By what amount are the clocks out of synchronisation in this frame?
c) Imagine instead that the clocks are aligned horizontally in the frame in which the whole rig is moving horizontally, as above. Note that the clocks are at rest relative to each other and assume they are synchronised in their own rest frame. By what amount are they out of synchronisation in the frame in which the rig is moving?
(solved by @TSny )
There is a perfectly rough horizontal table. This table is pretty wide (actually it is a plane) and it rotates about some vertical axis. Angular velocity is a given constant: ##\Omega\ne 0##. Somebody throws a homogeneous ball on the table. The ball has a mass ##m## and a radius ##r##.
The ball begins to roll on the table without slipping. Initially (##t=0##) a vector ##\boldsymbol{OP}## (see the picture) is known; initial angular velocity of the ball and initial velocity of its center of mass ##S## are also known vectors.
1) Prove that the center of the ball describes a circle relative to the lab frame;
2) Find the radius of this circle.
Problem 2 (@wrobel )
(solved by @TSny , @Chestermiller )
There are four homogeneous rods. Each rod has mass ##m## and length ##b##. The ends of the rods are connected by frictionless hinges such that the rods form a rhombic frame (see the picture). This frame is shaped as square ##ABCD## and put at rest on a smooth horizontal table. Then one applies a force ##F## to the hinge ##A## along the diagonal #AC#. Find acceleration of the point ##C## right after the force has been applied.
Problem 3 (@PeroK )
(solved by @Gaussian97 )
a) Suppose a particle has threevelocity ##\vec{u} = (u_x, u_y, u_z)## in frame ##S##, which is moving with velocity ##(v, 0, 0)## relative to frame ##S'##. Show that the gamma factor of the particle in frame ##S'## is given by:
$$\gamma'_u = \gamma_v \gamma_u(1 + \frac{vu_x}{c^2})$$
b) Imagine a rig, consisting of two vertical poles, a distance ##L## apart. Two synchronised clocks are mounted on the poles and moving upwards at speed ##u##, relative to the rig, remaining horizontally aligned with respect to the rig.
Imagine a frame of reference in which the whole rig is moving horizontally at speed ##v##. By what amount are the clocks out of synchronisation in this frame?
c) Imagine instead that the clocks are aligned horizontally in the frame in which the whole rig is moving horizontally, as above. Note that the clocks are at rest relative to each other and assume they are synchronised in their own rest frame. By what amount are they out of synchronisation in the frame in which the rig is moving?
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