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.

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.

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 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.

A uniform rubber band of mass $m$, natural radius $r_1$, and stiffness coefficient $k$ is placed horizontally on a vertical cylinder of radius $r_2$ ($r_2 > r_1$). The mass distribution of the band remains uniform. The coefficient of static friction between the band and the cylinder is $\mu$. The cylinder rotates about its vertical axis with angular velocity $\omega$.

An inclined plane with side ratios 3:4:5 is fixed on a horizontal turntable. A wooden block is placed on the inclined plane and remains stationary at a distance $r = 40$ cm from the center of the turntable. The coefficient of static friction between the block and the plane is $\mu_s = 1/4$.

A curved section of railway track has a radius of curvature $r$. The distance between the two rails is $L$, as shown in the figure.