Circular Motion

A small ball is tied to a 1 m long string and swung in a vertical circle.

A bolt is threaded onto one end of a thin horizontal rod, and the rod is then rotated horizontally about its other end. An engineer monitors the motion by flashing a strobe lamp onto the rod and bolt, adjusting the strobe rate until the bolt appears to be in the same eight places during each full rotation ofthe rod. The strobe rate is 2000 flashes per second; the bolt has mass 30 g and is at radius 3.5 cm.

A circular highway curve of radius $R$ is designed for a specific speed $v_d$ such that no friction is required to navigate the turn. A car travels along this curve at a different speed $v_a$ on a rainy day.

A puck of mass $m$ slides in a circle of radius $r$ on a frictionless table. The puck is attached to a hanging cylinder of mass $M$ by a cord that extends through a hole in the table.

Figure depicts an overhead view of a car's path as it travels toward a wall. Assume that the driver begins to brake the car when the distance to the wall is $d$, and take the car's mass as $m$, its initial speed as $v_0$, the coefficient of static friction as $\mu_s$, and the coefficient of kinetic friction as $\mu_k$. Assume that the car's weight is distributed evenly on the four wheels.

A ball of mass $m = 1.34$ kg is connected by two massless strings, each of length $L = 1.70$ m, to a vertical, rotating rod. The strings are tied to the rod with a vertical separation $d = 1.70$ m and are taut. The ball rotates in a horizontal circle at a constant speed. The tension in the upper string is $T_u =35$ N.

An object of mass $m$ is placed on the conical surface of an umbrella. The cone surface makes an angle $\theta$ with the horizontal. The umbrella rotates about its vertical axis with a constant angular velocity $\omega$. The object is at a horizontal distance $r$ from the axis of rotation.

A certain string can withstand a maximum tension of 40 N without breaking. A child ties a 0.37 kg stone to one end and, holding the other end, whirls the stone in a vertical circle of radius 0.91 m, slowly increasing the speed until the string breaks.

A block C of mass $m = 2$ kg is placed on a smooth inclined plane with an angle of inclination $\alpha = 30^\circ$. It is attached to a light spring with an unstretched length of $l_0 = 0.2$ m. The other end of the spring is fixed at the top of the incline. When the system is at rest, the spring stretches to a length of $l_1 = 0.25$ m. The entire system (incline and block) rotates about a vertical axis AB as shown. (Use $g = 9.8$ m/s²)

A car of mass $m$ is driving over an arched bridge. The top of the bridge can be approximated as a circular arc with radius $R$.