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7 problems tagged with Non-inertial reference
P0501
Advanced Mechanics › DynamicsPebble Slipping on a Rolling Wheel
A wheel of radius R rolls without slipping along a straight line on a horizontal surface at a constant speed v. A small pebble of mass m is gently placed on the top of the wheel. The coefficient of static friction between the pebble and the wheel is μ.
P0502
Advanced Mechanics › DynamicsMaximum Rotation of a Rubber Band on a Cylinder
A uniform rubber band of mass $m$, natural radius $r_1$, and stiffness coefficient $k$ is placed horizontally on a vertical cylinder of radius $r_2$ ($r_2 > r_1$). The mass distribution of the band remains uniform. The coefficient of static friction between the band and the cylinder is $\mu$. The cylinder rotates about its vertical axis with angular velocity $\omega$.
P0503
Advanced Mechanics › DynamicsTension in Double Pendulum After Collision
A double pendulum system consists of two inextensible light strings of lengths $l_1$ and $l_2$, and two small balls A and B with masses $m_1$ and $m_2$ respectively. Ball A is is connected with string $l_1$ from the ceiling, and ball B is at the bottom, connected to ball A via string $l_2$. The system is initially in a vertical equilibrium position. A third ball C, also with mass $m_1$, collides elastically with ball A at a horizontal velocity $v_0$.
- Find the tension $T_1$ in the upper string immediately after the collision.
- Find the tension $T_2$ in the lower string immediately after the collision.
P0504
Advanced Mechanics › DynamicsMotion of an Object on Perpendicular Belts
Two conveyor belts are in the same horizontal plane, moving perpendicularly to each other. Belt 1 moves at speed $v_1$, and belt 2 moves at speed $v_2$. An object of mass $m$ is delivered from belt 1 onto belt 2. The coefficient of kinetic friction between the object and the belts is $\mu$. The point where the object lands on belt 2 is the origin O of a stationary coordinate system, with belt 1's motion along the x-axis and belt 2's motion along the y-axis. The belts are sufficiently wide.
- Find the coordinates (x, y) of the object in the stationary frame when it stops sliding relative to belt 2.
- Find the total time the object slides on belt 2.
P0505
Advanced Mechanics › DynamicsBlock on a Rotating Inclined Plane
An inclined plane with side ratios 3:4:5 is fixed on a horizontal turntable. A wooden block is placed on the inclined plane and remains stationary at a distance $r = 40$ cm from the center of the turntable. The coefficient of static friction between the block and the plane is $\mu_s = 1/4$.
P0506
Advanced Mechanics › DynamicsBlock Sliding Down a Frictionless Movable Wedge
A wedge ABC of mass $M$ and height $h$ has an inclined surface AC with an angle $\theta$. A small object of mass $m$ is placed at the top A and slides down from rest. All contact surfaces are frictionless. Find:
- The displacement of $M$ when $m$ slides from the top to the bottom.
- The acceleration of $M$ relative to the ground, $a_1$, while $m$ is sliding.
- The acceleration of $m$ relative to $M$, $a'_2$.
- The acceleration of $m$ relative to the ground, $a_2$.
- The interaction force $N$ between $m$ and $M$.
- The normal force $R$ between $M$ and the table.
P0507
Advanced Mechanics › DynamicsTrain Dynamics on a Banked Curve
A curved section of railway track has a radius of curvature $r$. The distance between the two rails is $L$, as shown in the figure.
- When a train passes through this curve at a rated speed $v_0$, what should be the height difference $h$ between the outer and inner rails so that the rails experience no lateral thrust?
- When the train passes through the curve at a speed $v (v > v_0)$, to prevent it from overturning, what is the maximum possible height of its center of mass?
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