Newton's Law

A small ball is thrown vertically upwards. When it returns to the launching point, its speed is 3/4 of the initial speed. Assume the air resistance experienced by the ball during its motion is a constant force. Take $g = 10$ m/s².

As shown in the figure, three objects with masses $m_1$, $m_2$, and $m_3$ are hung in series and are initially at rest. Object 1 is attached above to an inextensible light string a. It is connected to object 2 by a light spring b, and object 2 is connected to object 3 by a light spring c. The springs are not necessarily identical.

As shown in the figure, for an ellipse with semi-major axis $a$ and semi-minor axis $b$, assume the major axis is vertical. A point mass of mass $m$ slides without friction under gravity along various straight chords passing through the center of the ellipse (referred to as "diameters").

As shown in the figure, the masses of the three objects A, B, and C are $m_A = \frac{2\sqrt{3}}{3}m$, $m_B = 2m$, and $m_C = m$, respectively. Object C is connected to sliders A and B by two ropes that pass over two light, smooth pulleys, $O_1$ and $O_2$, at the same height. Sliders A and B are on fixed, smooth inclined planes with angles 60° and 30°, respectively. The system is initially in equilibrium. An object D with mass $m_D=m$ is gently hung on a hook below object C.

As shown in the figure, a triangular wedge P of mass $M$ is placed between two blocks Q1 and Q2, of mass $m_1$ and $m_2$ respectively. The blocks can slide on a horizontal plane, and the system is released from rest. The angles between the sides of the wedge and the vertical are $\alpha$ and $\beta$. All surfaces are frictionless.

As shown in the figure, a small ball of mass $m$ is launched horizontally from point A with an initial velocity $v_0$. Under the influence of gravity and air resistance, it lands at point B. The velocity at B is $v'_0$, directed at an angle $\theta$ below the horizontal. The air resistance is given by the formula $f = -kv$, where $k$ is a positive constant and $v$ is the ball's velocity.

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:

1. One pulley is fixed to the wall (left).
2. One pulley is attached to the front of the block.
3. 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).

1. Part A: Ideal Pulleys: Assume the pulleys are massless and frictionless.
2. Part B: Real Pulleys: Assume both pulleys are solid disks with mass $M_p$, radius $R$, and moment of inertia $I$.