Circular Motion

A car travels along a circular track with an initial velocity of $v_0 = 7.0$ m/s and undergoes uniform deceleration. At time $t_1 = 5$ s, the angle between the total acceleration vector and the velocity vector is $\theta_1 = 135^\circ$. After an additional time of $t_2 = 3$ s (i.e., at a total time of $t_1 + t_2 = 8$ s), the angle between the total acceleration and velocity vectors becomes $\theta_2 = 150^\circ$.

On a smooth circular tabletop of radius $R$ with its center at O, a vertical post is fixed at the center. The intersection of the post with the tabletop is a convex, smooth, closed curve C. An inextensible, flexible, light string has one end fixed at a point on the curve C, and the other end is attached to a small block of mass $m$. The block is placed on the tabletop, and the string is pulled taut. The block is then given an initial velocity of magnitude $v_0$ in a direction perpendicular to the string. As the block moves on the tabletop, the string wraps around the post. The string breaks when its tension reaches $T_0$. It is assumed that the initial tension $T$ is less than $T_0$, and the block always remains on the tabletop before the string breaks.

As shown in the figure, a semi-cylinder with radius R undergoes uniformly accelerated motion with a constant horizontal acceleration $a$. A vertical rod, constrained to move only in the vertical direction, rests on the curved surface of the semi-cylinder. At the instant when the semi-cylinder's horizontal velocity is $v$, the contact point P between the rod and the semi-cylinder is at an angular position $\theta$ with respect to the vertical axis.

A spool is formed by a small cylinder of radius $r$ coaxially fixed to a large cylinder of radius $R$. A string is wound on the inner cylinder. The string is drawn out, passes over a pulley Q, and its end is pulled with a constant velocity $v$. Simultaneously, the spool rolls without slipping on a horizontal surface. The segment of the string from the spool to the pulley, PQ, makes an angle $\varphi$ with the horizontal, as shown in the diagram.

Two small balls are undergoing uniform circular motion.

A particle undergoes uniform circular motion. At time $t_1$, its velocity is $\vec{v}_1 = v_x \hat{i} + v_y \hat{j}$. At a later time $t_2$, its velocity is $\vec{v}_2 = -\vec{v}_1$. The time interval is $\Delta t = t_2 - t_1$.

An object is in uniform circular motion (UCM).

As shown in the figure, one end of a thin rope is fixed at point A. A weight B is attached to the rope at a distance $a$ from A, making the length of segment AB constant, $l_{AB} = a$. The other end of the rope passes over a fixed pulley at point C. Points A and C lie on the same horizontal line. The free end of the rope is pulled with a constant speed $v$. At the instant shown, the rope segments AB and BC make angles $\alpha$ and $\beta$ with the horizontal, respectively.

A particle undergoes uniform circular motion. At a given instant, its position vector is $\vec{r}$, its velocity is $\vec{v}$, and its acceleration is $\vec{a}$.

Due to the Earth's rotation, objects on its surface undergo uniform circular motion around the axis of rotation. Consider two objects, one located in Beijing (at 40° N latitude) and one at the equator.