A body of mass m2 has one of its surfaces in the form of a...

A body of mass $\mathrm{m}_{2}$ has one of its surfaces in the form of a quarter circle of radius $\mathrm{R}$. A mass $\mathrm{m}_{1}$ is placed on top of a plunger assembly. Assume all surfaces are frictionless. The final velocity of $m_{2}$, if plunger starts to descend from initial position $B$ as shown, is (a) $\sqrt{2 \mathrm{gR} \cdot \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}}$ (b) $\sqrt{2 \mathrm{gR}(1-\sin \theta) \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}}$ (c) $\sqrt{2 g R(1-\cos \theta) \frac{m_{1}}{m_{2}}}$ (d) $\sqrt{2 g R \cdot \frac{m_{2}}{m_{1}}}$

A body of mass $\mathrm{m}_{2}$ has one of its surfaces in the form of a quarter circle of radius $\mathrm{R}$. A mass $\mathrm{m}_{1}$ is placed on top of a plunger assembly. Assume all surfaces are frictionless. The final velocity of $m_{2}$, if plunger starts to descend from initial position $B$ as shown, is
(a) $\sqrt{2 \mathrm{gR} \cdot \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}}$
(b) $\sqrt{2 \mathrm{gR}(1-\sin \theta) \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}}$
(c) $\sqrt{2 g R(1-\cos \theta) \frac{m_{1}}{m_{2}}}$
(d) $\sqrt{2 g R \cdot \frac{m_{2}}{m_{1}}}$

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Key Concepts

Motion on Curved Surfaces

When a body moves along a curved (or circular) path, like a quarter circle, the geometry of the path influences the dynamics of the motion. The curvature introduces factors such as the effective change in height and the corresponding gravitational energy conversion, which are central to solving for the system's final speed.

Conservation of Energy

This principle states that in an isolated, frictionless system, the total energy remains constant. When applied to mechanics, the gravitational potential energy lost by one mass is converted into the kinetic energy of another, allowing the determination of velocities and other motion characteristics.

Constraint Relationships in Mechanical Systems

In systems where bodies interact through connecting surfaces or mechanisms, the motion of one body can restrict or dictate the movement of another. Understanding these constraints, which often come from geometric or kinematic relations, is essential to relate displacements and velocities among the bodies.

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