A bug flying horizontally at 1.0 m / s collides and sticks to the end of a uniform stick hanging

step 1 So the situation over here is there is a stick of length L, a bug flying with a velocity of V. I

A bug flying horizontally at 1.0 m / s collides and sticks...

A bug flying horizontally at $1.0 \mathrm{m} / \mathrm{s}$ collides and sticks to the end of a uniform stick hanging vertically. After the impact, the stick swings out to a maximum angle of $5.0^{\circ}$ from the vertical before rotating back. If the mass of the stick is 10 times that of the bug, calculate the length of the stick.

Key Concepts

Inelastic Collisions in Rotational Dynamics

In an inelastic collision, objects stick together after impact and kinetic energy is not conserved due to energy losses (usually as heat or deformation), but momentum—specifically angular momentum in rotational cases—is conserved. This concept helps analyze the subsequent rotational motion resulting from the collision by providing a way to calculate post-collision velocities or angular speeds despite the loss of energy.

Moment of Inertia

The moment of inertia quantifies how mass is distributed relative to the axis of rotation and determines an object's resistance to changes in its rotational state. For extended bodies, such as a uniform stick, the moment of inertia depends on both the mass of the object and its length, and it plays a key role in linking angular momentum and rotational kinetic energy in dynamics problems.

Conservation of Angular Momentum

In rotational dynamics, the conservation of angular momentum states that if no external torque acts on a system, the total angular momentum before an event (such as a collision) must equal the total angular momentum after the event. This principle is crucial for analyzing impacts or collisions where objects start rotating, even if the collision is inelastic, because it allows us to relate the initial linear momentum of a moving body to the rotational motion acquired by the system.

Energy Conservation in Rotational Systems

After an inelastic collision causes a system to rotate, the rotational kinetic energy is converted into gravitational potential energy as the system swings upward (like in a pendulum). By equating the initial rotational kinetic energy to the potential energy at the maximum angular displacement (ignoring friction and air resistance), one can determine important parameters of the system such as the length of a pendulum or stick.

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