A uniform solid cylinder of mass M and radius R can freely rotate around its axis O. And there is a spring of relaxed length L and stiffness K attached to the cylinder and a static wall. See picture below.
Initially, the spring is relaxed. As the cylinder starts rotating, the spring will wind the cylinder. The surface of cylinder is very rough, so that the spring does not slip with the cylinder's surface.
Find the minimum initial angular speed of the cylinder, ω0, so that it can rotate to angle 2π!
Click here to read the solution.
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Showing posts with label ***. Show all posts
A complex circuit consists of some resistors. An ideal current source is then connected through two points at the circuit. Then an ideal voltmeter is connected to two nodes of a resistor in the circuit. Reading of the voltmeter is V.
Then the voltmeter is substituted with an ideal ammeter. The reading is I.
If the current source is then disconnected to the circuit and the ammeter is substituted with ohmmeter, prove that reading of the ohmmeter is V/I!
Click here to read the solution.
Consider the same case from previous problem. But there is an external force works on the block, and bring the block up to a height of 2h very slowly.
Find the final average velocity of the ball!
Click here to read the solution.
In picture above, a small ball of mass m moves with average velocity v vertically up and down due to collisions with a block of mass M and a floor. The block is floating in the air due to the collisions with the ball.
Find h, the average height of the block!
Gravitational acceleration g. Assume and
.
Click here to read the solution.
A uniform flat square is placed on a rough floor. If the square is rotated around its edge, it takes time t to completely stop.
Calculate the time it takes ,t', to completely stop if it's rotated around its center with same initial angular velocity!
Click here to read the solution.
A ball of mass m is hang by a string of length l on a rough inclined plane with coefficient of friction μ and angle θ with the horizontal (μ=tanθ). Initially the ball is at the lowest position.
Calculate the minimum initial velocity, v, should be given to the ball to make a full circle of its path!
Click here to read the solution.
Two identical small balls are shown in picture above. A ball is thrown from the ground and another ball is dropped without any initial velocity.
First, consider no air friction. In order to make the balls collide each other, the ball on ground has to make an angle α from the ground, where tan(α)=h/d.
Then consider a more real case, there's air friction which is proportional to ball's velocity, F=-kv (see previous problem). In order to make the balls collide, determine whether angle of the ball has to be greater than α, less than α, or equals to α!
A rectangle block with dimension a*b*c floats on water. The block's mass density is a half of water density, so only half part of the block inside the water. See picture below.
Find the minimum value of a/b in order to make it stable, if it's rotated slightly around axis O!
(Hint: if you understand the concept, it would be much easier)
A cylinder of radius r and mass m move with speed v on a horizontal plane. In front of it, there is a block with height h (h<r) and mass M rest on the same plane.
The cylinder then hit the block inelastically and start moving up onto the block. Ignore all friction.
Determine minimum velocity of the cylinder so that it can go up to top of the block!
An old bridge can be modeled as a half-cylinder's shell. Consider an old bridge of radius r is placed on a ground. The ground is not perfectly horizontal, but it makes an angle θ with the horizontal. Then a mass M is placed on top of the bridge. See picture below.
The bridge's mass can be ignored, compared to M. Assume that the ground is frictionless and the bridge is not glued on the ground. Determine the position of a point where the bridge is most likely to break!
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