Mechanical Energy And Conservation

A massless rigid rod of length $L$ has a ball of mass $m$ attached to one end. The other end is pivoted so that the ball moves in a vertical circle. First, assume there is no friction at the pivot. The system is launched downward from the horizontal position A with an initial speed $v_0$. The ball just barely reaches point D (the top) and then stops.

A small block is sent through point A with a speed of $v_A = 7.0$ m/s. Its path is frictionless until it reaches point C. The section from C to D has length $L = 12$ m and a coefficient of kinetic friction $\mu_k = 0.70$. The heights are $h_1 = 6.0$ m and $h_2 = 2.0$ m, relative to the lowest point B.

A block is released from rest at a height $d=40$ cm. It slides down a frictionless ramp onto a first plateau of length $d$ with a coefficient of kinetic friction $\mu_k = 0.50$. If still moving, it descends a second frictionless ramp through a height $d/2$ and travels across a second plateau of length $d/2$, where the coefficient of kinetic friction is also $0.50$. Finally, if the block is still in motion, it moves up a frictionless ramp until it momentarily stops.

A particle can slide along a track with elevated ends and a flat central part, as shown in Figure. The flat part has length $L = 40$ cm. The curved portions of the track are frictionless, but for the flat part the coefficient of kinetic friction is $\mu_k = 0.20$. The particle is released from rest at point A, which is at height $h = L/2$.

A boy is initially seated on the top of a frictionless hemispherical ice mound of radius $R$. He begins to slide down the ice with a negligible initial speed.

An object with a mass of $m = 1$ kg is in free fall. Take the acceleration due to gravity as $g = 10$ m/s².

A projectile is launched and follows a trajectory, eventually returning to its original launch height.

A uniform cord of length $L$ and mass $M$ is initially stuck horizontally to a ceiling. Later, it hangs vertically from the ceiling with one end still stuck.

A force of $F = 120$ N is required to compress a spring by $x_1 = 4$ cm.

A spring with a natural length of $20$ cm and a spring constant of $k = 1000$ N/m is stretched.