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15 problems tagged with rigid body dynamics
P0647
Intermediate Mechanics › Rotational MotionTorque on a System with Constant Acceleration
An electric motor drives a system with a moment of inertia $I = 50 \text{ kg} \cdot \text{m}^2$ in fixed-axis rotation. The system starts from rest and, with uniform acceleration, reaches a rotational speed of $f = 2 \text{ r/s}$ in $t = 0.5 \text{ s}$.
P0666
Intermediate Mechanics › Rotational MotionBall rolls down slope without slipping
A ball of radius $r$ and mass $m$ rolls down a slope with incline angle $\theta$ with slipping. The initial verticle height of the ball is $h$ from the bottom of the slope.
- Can this happen on the frictionless surface. If not what is minimum static friction coefficient? And what is the work done by the friction?
- What is the final speed $v$ of the ball, and how long does it take to get to the bottom? Solve it in two ways: kinemetics and energy conservation.
- Compare the final speed for a ball, ring and a disk.
P0720
Intermediate Mechanics › Rotational MotionThe "Self-Propelled" Man
A block of mass $M$ rests on a smooth, frictionless horizontal floor. A man of mass $m$ stands on top of the block. A system of two pulleys is used to accelerate the block:
- One pulley is fixed to the wall (left).
- One pulley is attached to the front of the block.
- A massless, inextensible rope is anchored to the wall, passes over the block's pulley, then back over the wall's pulley, and finally into the hands of the man.
The rope segments are all horizontal. The coefficient of static friction between the man's shoes and the top of the block is $\mu_s$. The man pulls the rope with a tension force $T$, causing the entire system (block + man) to accelerate towards the wall (to the left).
- Part A: Ideal Pulleys: Assume the pulleys are massless and frictionless.
- Part B: Real Pulleys: Assume both pulleys are solid disks with mass $M_p$, radius $R$, and moment of inertia $I$.
- Kinematic Constraint: If the man pulls a length $L$ of rope through his hands relative to his position on the block, what is the displacement $d$ of the block relative to the ground?
- Ideal Dynamics (Part A): Determine the acceleration $a$ of the system (man + block) in terms of $T$, $m$, and $M$.
- The "Slippery Shoes" Limit (Part A): Determine the maximum tension $T_{max}$ the man can apply before his shoes slip on the block. Discuss how the direction of the required friction force depends on the ratio of the masses $M$ and $m$.
- Kinetic Energy (Part A): What is final kinetic energy of the system after the block has moved a distance $d$ from rest?
- Non-Ideal Dynamics(Part B): Derive the new acceleration $a'$ of the system, taking into account the moment of inertia $I$ of both pulleys. Express your answer in terms of $T$, $m$, $M$, $I$, and $R$.
- Friction Comparison(Part B): Does the presence of heavy pulleys (non-zero $I$) increase or decrease the static friction force required between the man's shoes and the block to prevent slipping (compared to the ideal case in Q3)? Justify your answer physically.
P0722
Advanced Mechanics › Rotational MotionFalling stick against the wall
A uniform stick of mass $m$ and length $l$ leans against a smooth wall vertically. The stick can slide against the wall and the ground frictionlessly. With small perturbation, the stick starts to fall.
- Find at which point $\theta$ the stick starts to leave the wall.
- Find the angular and translational velocity as a function of angle $\theta$ (between 0 and 90 deg) between the floor and the stick.
- Find the distance between the center of the mass and the wall when the stick falls to the ground.
P0747
Advanced Mechanics › Rotational MotionStick sliding down the wall
A uniform stick AB of length $2l$ is leaning against a wall, with the end A on the wall and B on the ground. A ball of mass $m$ is fixed at the center of the stick. As the stick slides down with the end moving at constant velocity $v$, when the angle between the stick and the ground is at $45 \deg$, what is the force of the ball applied on the stick?
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