Mechanical Energy And Conservation

As shown in the figure, two fixed pulleys are separated by a distance $2d$. Two small balls A and B of equal mass are connected by a string that passes over the pulleys and supports a third ball C. As A and B descend with velocity $v$, ball C is pulled upward. At a certain moment, the angle between the two string segments connected to ball C is $2\alpha$. Neglect the mass of the string and pulleys, and any friction.

A smooth, thin circular tube of radius R is oriented in a vertical plane. Two small balls, A and B, with masses $m_1$ and $m_2$ respectively, move inside the tube. Both balls have a speed of $v_0$ when they pass the lowest point of the tube.

Figure shows a plot of potential energy $U$ versus position $x$ of a 0.90 kg particle that can travel only along an x axis. (Nonconservative forces are not involved.) Three values are $U_A = 15.0$ J, $U_B = 35.0$ J, and $U_C = 45.0$ J. The particle is released at $x = 4.5$ m with an initial speed of 7.0 m/s, headed in the negative x direction.

A skier weighing 600 N goes over a frictionless circular hill of radius $R = 20$ m. Assume that the effects of air resistance on the skier are negligible. As she comes up the hill, her speed is 8.0 m/s at point $B$, at an angle $\theta = 20^\circ$.