FIGURE P2.64 shows a fixed vertical disk of radius R. A thin, frictionless rod is attached to the bottom point of the disk and to a point on the edge, making angle Φ (Greek phi) with the vertical. Find an expression for the time it takes a bead to slide from the top end of the rod to the bottom.

Verified step by step guidance

1. Step 1: Analyze the setup. The bead slides along a frictionless rod that is inclined at an angle Φ with the vertical. The rod is attached to the bottom of a fixed vertical disk of radius R. The bead starts at the top end of the rod and slides to the bottom. The motion is governed by gravitational acceleration and the geometry of the rod.
2. Step 2: Determine the forces acting on the bead. The bead experiences a gravitational force, which can be decomposed into two components: one parallel to the rod (responsible for the bead's acceleration) and one perpendicular to the rod (which is counteracted by the normal force from the rod). The parallel component of the gravitational force is given by m * g * sin(Φ), where m is the mass of the bead, g is the acceleration due to gravity, and Φ is the angle of inclination.
3. Step 3: Use Newton's second law to find the acceleration of the bead along the rod. The net force along the rod is m * g * sin(Φ), and the acceleration a is given by a = g * sin(Φ). This acceleration is constant since the rod is frictionless and the angle Φ does not change.
4. Step 4: Relate the length of the rod to the geometry of the disk. The rod forms a chord of the circle, and its length can be expressed in terms of the radius R and the angle Φ. Using trigonometry, the length of the rod L is given by L = R * sin(Φ).
5. Step 5: Use kinematic equations to find the time it takes for the bead to slide down the rod. The bead starts from rest, so the initial velocity v₀ = 0. The kinematic equation for constant acceleration is L = (1/2) * a * t², where L is the length of the rod, a is the acceleration, and t is the time. Substitute L = R * sin(Φ) and a = g * sin(Φ) into the equation to solve for t: t = sqrt(2 * R * sin(Φ) / g * sin(Φ)).

Key Concepts

Inclined Plane Dynamics

An inclined plane is a flat surface tilted at an angle to the horizontal. When an object slides down an inclined plane, its motion is influenced by gravitational force, which can be resolved into components parallel and perpendicular to the surface. The component of gravity acting parallel to the incline causes the object to accelerate down the slope, while the perpendicular component affects the normal force.

Kinematics of Motion

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. Key equations relate displacement, velocity, acceleration, and time. For an object sliding down an incline, the time taken to travel a certain distance can be calculated using these equations, factoring in the initial conditions and the acceleration due to gravity.

Energy Conservation

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the bead sliding down the rod, gravitational potential energy is converted into kinetic energy as the bead descends. This relationship can be used to derive expressions for the bead's speed and the time taken to slide down the incline.