Newton's Law

As shown in Figure, a smooth cylinder of mass $m$ is placed in a V-shaped groove of mass $M$. The cross-section of the groove is symmetric, with each side making an angle $\alpha$ with the horizontal, where $\alpha < 45^\circ$. The groove rests on a frictionless horizontal surface. The groove is connected by a thin, massless string that passes over a frictionless, massless pulley to a hanging weight of mass $m_1$. The system is released from rest. When $m_1$ is sufficiently large, the cylinder can leave the groove.

On a wire that is at an angle $\alpha$ with horizonal, there is a ring of mass $m_1$ that can slide without friction o the wire. Another ring of mass $m_2$ is connected to ring $m_1$ by a massless string AB. At the beginning, the system is at rest, and the string AB is at vertical position.

Three connected blocks with masses $m_1$, $m_2$, and $m_3$ are pulled to the right on a horizontal frictionless table by a force of magnitude $T_3$.

Three blocks with masses $m_A$, $m_B$, and $m_C$ are connected by massless cords that loop over frictionless pulleys as shown. Block B lies on a frictionless table. The blocks are released from rest. Assume $m_C > m_A$.

Three blocks with masses $m_A$, $m_B$, and $m_C$ are connected by massless cords as shown. One cord passes over a massless, frictionless pulley. The block with mass $m_A$ rests on a frictionless horizontal surface. The system is released from rest.

Four objects with masses $m_1, m_2, m_3,$ and $m_4$ are connected by three massless cords and pulled along a frictionless horizontal surface. The tension in the cord pulling the entire system is $T_4$, and the tension in the cord between objects 2 and 3 is $T_2$. The masses $m_1, m_3, m_4$, and the tensions $T_2, T_4$ are known.

A horse pulls a barge of mass $m$ with a force of magnitude $F_H$ at an angle $\theta$ relative to the direction of motion. The barge moves along the positive x-axis with a constant acceleration of magnitude $a$.

A constant horizontal force $\vec{F}_a$ is applied to a system of two blocks, A and B, with masses $m_A$ and $m_B$ and total mass $M = m_A + m_B$. Case 1: When the force is applied to block A, the magnitude of the contact force between the blocks is $F_1$. Case 2: When the same force is applied to block B, the magnitude of the contact force is $F_2$.

Two blocks of mass $m_1$ and $m_2$ are connected by a massless cord passing over a frictionless, massless pulley, forming an Atwood's machine. Assume $m_2 > m_1$.

A man of mass $m_1$ lowers himself to the ground from a height $h$ by holding a rope that runs over a frictionless pulley to a sandbag of mass $m_2$. The system starts from rest, and we assume $m_1 > m_2$.