A scooter of length l has two wheels. The scooter moves with CM's speed v on a rough plane and is going to turn its direction without change its speed. To turn its direction, the front wheels is rotated to an angle θ.
Calculate the minimum coefficient of friction between the plane and the wheels, μ, so that the scooter does not slip to the plane! Assume CM of the scooter is at the middle.


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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!


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A uniform rod of mass M and length l is placed on a rough floor and a rough wall. The rod makes an angle θ to the floor.What is the minimum coefficient of friction of the floor and the wall in order to prevent the rod from slipping? Assume the floor and the wall made by the same material.


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Why is it much easier to break a ruler by bending it than by pulling it?


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A small ball of mass M is hung with a spring of stiffness k and zero relaxed length. The ball is in gravitational field, g. Then the ball is brought down to a point (x0, y0) as shown in picture below, and then released.How will path of the ball? And also find coordinate of the ball when it reaches its highest position!


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A track consists of two rough planes forming an angle 2θ to each other. It's symmetric to vertical axis and makes an angle φ to the horizontal.
Coefficient of friction of the planes are different. The left side is μ0 and the right side is μ.
Then a ball of radius R and mass M (moment inertia I=ηMR^2) is placed on the track.


As the ball moves, it slips with left side of the track and does not slip with right side of the track.
Find the minimum value of μ! (in μ0, φ, and θ)


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Consider two rods forming V-shape with angle θ and a ball, of radius r, mass m, and moment inertia I, on it. See picture below.
The ball is released at position l from rods' adjacent and start moving because of gravitational acceleration, g. The rods are rough enough, so it prevents the ball from slipping in its movement.
Calculate the initial angular acceleration of the ball!


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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!


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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 .


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A block of mass m is oscillating with a spring of stiffness k. See the picture.In order to increase its amplitude, one of end of the spring (point A) is suddenly moved with a velocity v to the left and kept moving with that velocity.
Determine when it should be suddenly moved in order to maximize its amplitude!
How's the result if it's not suddenly moved, but accelerated by a to the left!


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Two small balls of mass m are hung by strings of length l to a static nail. The balls are connected with a massless rod of same length, l. Initially, the balls are held at position like shown below.The balls are then released and start moving down.Calculate the maximum tension, T, of a string during the motion!

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A pencil can be modeled as a hexagonal prism (in this case). When a pencil rolls on a rough floor, it will rotate around one of its edge, and then the next edge will hit the floor, and then the pencil will continue rotating with the edge as a new rotation axis. To simplify the problem, assume mass of the pencil is concentrated at its center.Calculate the minimum value of coefficient of friction between the pencil and the floor so that the pencil doesn't slip when its edge hits the floor!
Assume the contact between the pencil and floor only occurs at the pencil's edge.


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A uniform rod of length l and mass m is hung by two strings of length l at its two ends, like picture below. Then the rod is rotated to an angle θ and released without any initial velocity or angular velocity.
Calculate the magnitude of acceleration, a, experienced at its end (point A) immediately after it's released!
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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!
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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!


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A small ball of mass m is hang by a string of length l and attached to a spring of stiffness k like shown below.The spring can move freely in vertical direction, but its orientation is always horizontal. The ball is in equilibrium when it forms an angle θ with horizontal.
Determine period of oscillation, T, if the ball move slightly on xy-plane!
Gravity acceleration g.

A cylinder of radius R and mass M is on a rough inclined plane with inclined angle θ from horizontal. Let μ be the coefficient of friction between the inclined plane and the cylinder. The cylinder makes an angle φ with the generatrix (i.e. the line of intersection of the plane and the horizontal surface). Gravity acceleration is g.
Determine minimum value of μ in order to prevent the cylinder from slipping!

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!

Two identical ball are placed at same height from ground. Ball A is thrown horizontally and ball B is dropped without any initial velocity. The balls experience air friction which is proportional to ball's velocity, F=-kv, where k is a constant and minus sign means the direction of force is opposite to direction of velocity.
Determine which ball hits the ground first!

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!