Mechanical Energy And Conservation

As shown in Figure below, a small block of mass m is released from rest at height H on a smooth track. The track becomes a vertical circular loop of radius R at the bottom. Neglect any resistance.

A jar of mass $m$ moves up an incline of angle $\theta$. At a distance $d_0$ from the bottom, its speed is $v_0$. The coefficient of kinetic friction is $\mu_k$.

A stone of weight $W$ is launched vertically up from the ground with an initial speed $v_0$. It experiences a constant air drag force $F_d$ throughout its flight.

A bundle of mass $m$ starts up an incline of angle $\theta$ with an initial kinetic energy $K_0$. The coefficient of kinetic friction between the bundle and the incline is $\mu_k$.

A block of mass $m$ has an initial speed $v_A$ on a horizontal, frictionless surface. It moves up a frictionless curved ramp to a height $h$, where it encounters a rough section of length $L$ on an incline of angle $\theta$. The coefficient of kinetic friction on this section is $\mu_k$. Point B marks the end of the rough section.

A playground slide is in the form of an arc of a circle with radius $R$. A child of mass $m$ starts from rest at the top of the slide, which is at a vertical height $h$ above the bottom. The child's speed at the bottom is $v$.

As shown in the figure, a fixed pulley is installed on the edge of a smooth, fixed semicircular track of radius $R$. Two objects with masses $m_1$ and $m_2$ ($m_1 > m_2$) are connected by a light string passing over the pulley. The system is released from rest with $m_1$ at the top edge of the track, level with the pulley. It is known that $m_1$ can slide along the arc to the lowest point.

As shown, a block of mass $m$ is on a horizontal surface with a coefficient of kinetic friction $\mu_k$. It is pushed by a compressed spring with spring constant $k$. After leaving the spring (at the spring's equilibrium position), the block slides a distance $d$ before stopping.

An object of mass $m$ has an initial kinetic energy $K$. It moves up an inclined plane with an angle of inclination $\theta$.

At point O, at a height $H=2$ m above the ground, a thin string of length $l=0.75$ m is fixed. A small ball of mass $m=0.1$ kg is attached to the end of the string. The maximum tension the string can withstand is $T_m = 15.0$ N. The string is held horizontally and the ball is given an initial vertically downward velocity $v_0 = 8$ m/s. When the string swings to the vertical position, it hits a nail at O' directly below O, where $OO' = l' = 0.5$ m.