Circular Motion

Four children, A, B, C, and D, are at the vertices of a square, playing a chase game at the same constant speed $v$. A chases B, B chases C, C chases D, and D chases A. Each child always moves directly towards their target. At a certain instant, the four children form a square of side length $l$.

A rigid ring with radius $R$ undergoes pure rolling on a rigid horizontal surface. The center of the ring moves forward horizontally with a constant velocity $v_0$. Consider a point P on the ring that is at the same height as the center.

A rabbit runs along a straight line at a constant speed of $v_r = 5$ m/s. At a certain moment, a fox spots the rabbit and begins to chase it. The fox maintains a constant speed of $v_f = 4$ m/s, and its velocity vector at any instant points directly towards the rabbit. The distance between them initially decreases and then increases. The minimum distance between the fox and the rabbit is $d_{min} = 30$ m.

Two small balls, A and B, both undergo uniform circular motion. The ratio of their radii is $r_A : r_B = 1:2$. In the same time interval, ball A completes 75 revolutions while ball B completes 45 revolutions.

A particle undergoes uniform circular motion with speed $v$ over a horizontal xy coordinate system. At time $t_1$, its position is $\vec{r}_1 = x_1\hat{i} + y_1\hat{j}$, its velocity is $\vec{v}_1 = v\hat{j}$, and its acceleration is in the positive x-direction. At a later time $t_2$, its velocity is $\vec{v}_2 = -v\hat{i}$, and its acceleration is in the positive y-direction. The time interval $\Delta t = t_2 - t_1$ is less than one period.

A particle moves along a circular orbit, starting from rest and undergoing uniformly accelerated circular motion. The angle between the particle's total acceleration vector and its velocity vector is denoted by $\alpha$. The central angle corresponding to the arc traversed by the particle is denoted by $\theta$.

A particle undergoes counter-clockwise uniform circular motion. At time $t_1$, its acceleration is $\vec{a}_1 = a_{1x}\hat{i} + a_{1y}\hat{j}$. At a later time $t_2$, its acceleration is $\vec{a}_2 = a_{2x}\hat{i} + a_{2y}\hat{j}$. The time interval $\Delta t = t_2 - t_1$ is less than one period.

The relationship between linear velocity $v$, angular velocity $\omega$, and radius $r$ is given by a standard formula.

When asked about the relationship between the magnitude of centripetal acceleration ($a_c$) and the radius ($r$) in uniform circular motion, two students give different answers. Student 1 thinks of the formula $a_c = v^2/r$ and states that the magnitude of acceleration is inversely proportional to the radius. Student 2 thinks of the formula $a_c = \omega^2 r$ and states that the magnitude of acceleration is directly proportional to the radius.

A boy whirls a stone in a horizontal circle of radius $r$ and at height $h$ above level ground. The string breaks, and the stone flies off horizontally. It strikes the ground after traveling a horizontal distance of $d$.