Circular Motion

A circular curve of highway is designed for traffic moving at 60 km/h. Assume the traffic consists of cars without negative lift.

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.

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 $\theta_2 = 150^\circ$.

A heavy ball is tied to one end of a string, and the other end is held. The ball rotates rapidly in a horizontal plane, forming a conical pendulum.

A small iron block is placed on a horizontal turntable that rotates uniformly about a vertical axis. The block is 0.3 m from the center, and the coefficient of static friction between the block and the turntable is 0.4.

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 smooth, thin horizontal rod OA can rotate about a vertical axis MN passing through O. Two identical small objects, each with mass $m_1 = m_2 = m$, are threaded onto the rod. They are connected by two identical light springs, each with spring constant $k$ and natural length $L_0$. The first spring connects the axis at O to mass $m_1$, and the second spring connects mass $m_1$ to mass $m_2$.

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.

A wheel of radius R rolls without slipping along a straight line on a horizontal surface at a constant speed v. A small pebble of mass m is gently placed on the top of the wheel. The coefficient of static friction between the pebble and the wheel is μ.