(III) A long vertical hollow tube with an inner diameter of 1.00 cm is filled with SAE 10 motor oil. A 0.900-cm-diameter, 30.0-cm-long 150-g rod is dropped vertically through the oil in the tube. What is the maximum speed attained by the rod as it falls?

Verified step by step guidance

1. Determine the forces acting on the rod: The forces include the gravitational force pulling the rod downward, the buoyant force due to the displaced oil, and the drag force opposing the motion of the rod. The rod will reach terminal velocity when the net force is zero, meaning the downward gravitational force is balanced by the upward buoyant and drag forces.
2. Calculate the gravitational force on the rod: Use the formula $F$_g=mg, where $m$ is the mass of the rod (150 g or 0.150 kg) and $g$ is the acceleration due to gravity (9.8 m/s²).
3. Calculate the buoyant force: Use Archimedes' principle, which states that the buoyant force is equal to the weight of the displaced fluid. First, calculate the volume of the rod using its diameter and length, then multiply the volume by the density of the oil and $g$ to find the buoyant force.
4. Express the drag force: The drag force is given by the formula $F$_d=12C_dAρv², where $C$_d is the drag coefficient, $A$ is the cross-sectional area of the rod, $\rho$ is the density of the oil, and $v$ is the velocity of the rod. At terminal velocity, the drag force will balance the net downward force.
5. Set up the equilibrium condition: At terminal velocity, the gravitational force is equal to the sum of the buoyant force and the drag force. Write the equation $F$_g=F_b+F_d. Substitute the expressions for each force and solve for the terminal velocity $v$.

Key Concepts

Viscosity

Viscosity is a measure of a fluid's resistance to flow and deformation. In this scenario, the SAE 10 motor oil's viscosity affects how the rod moves through it, determining the drag force acting on the rod as it falls. Higher viscosity results in greater resistance, which influences the terminal velocity of the falling object.

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration. For the rod in the oil, terminal velocity occurs when the gravitational force is balanced by the drag force due to the oil's viscosity, leading to a steady speed.

Drag Force

Drag force is the resistance force experienced by an object moving through a fluid, which depends on the object's speed, shape, and the fluid's properties. In this case, the drag force acting on the rod as it falls through the motor oil is crucial for calculating the maximum speed, as it counteracts the gravitational force until equilibrium is reached.