AP Physics Laboratory #6.1: Analyzing Terminal Velocity Using an Interesting Version of Atwood s Machine Name: Date: Lab Partners: PURPOSE The purpose of this Laboratory is to study a system as it approaches terminal velocity both experimentally and mathematically. EQUIPMENT NEEDED Two solid spheres of identical mass, standard Atwood s Machine apparatus, long PVC pipe or transparent tube filled with water at STP. PROCEDURE Experimental Set-Up Set up the apparatus as illustrated in Figure 1, using identical objects for M 1 and M. Incorporate the PASCO data collecting system with graphs of position, velocity, and acceleration versus time. M M 1 Figure 1: New Scenario for Atwood s Machine Data Collection: 1. Position the masses such that the LED on the smart pulley is off.. Begin collecting data and release the masses. 3. Terminate data collection just prior to the falling sphere hitting the floor. 4. Use Data Studio to create graphs of position, velocity, and acceleration vs time. DATA 1. Spherical Objects: mass (m) = ; radius (R) = ;. Water (STP): density (ρ) = 1000.0 kg/m 3 ; viscosity (μ) = 0.000898 kg/m sec; 3. Graphs of position, velocity, and acceleration versus time.
THEORY Free Fall: Objects undergoing free-fall in the absence of air resistance accelerate downward at the local gravitational acceleration as a result of the influence of the gravitational force acting on them: F ma W mg ma a g Objects falling through a fluid also experience a drag force, who s magnitude is related to the density of the fluid (ρ), the frontal area (A), the drag coefficient (C d ), and the speed of the object (V) by the equation: 1 D AC dv Hence, when Newton s Second Law is applied in the downward direction, it is apparent that the object will accelerate so long as the weight exceeds the drag force: 1 F ma W D mg AC dv ma The maximum speed is reached when the object reaches value that the drag force is equal to the weight. Under these conditions, the acceleration is zero and the object travels downward at a terminal velocity with a magnitude of: v term mg AC d This analysis ignores the buoyancy force acting on the object. Recalling Archimedes Principle: The upward buoyant force (B) on an object is equal to the weight of the fluid it displaces. For a solid object falling through air, the buoyant force is very small (hence it is typically neglected). However, the same is not true if the object is falling or rising through water. Atwood s Machine: A desktop version of Atwood s Machine was used during Lab.1. In that situation, it was assumed that both the drag force and the buoyant force could be neglected. The resulting acceleration was:
New Scenario for Atwood s Machine: To enhance the drag force in an effort to yield a scenario for which terminal velocity can be readily measured in the classroom, the rising object will travel through water. This will result in significant drag and buoyant forces: M M 1 Figure 1 (repeated): New Scenario for Atwood s Machine 1. Derive an equation for the acceleration of the two masses in terms of parameters that can be measured (speed included) and fundamental constants (C d included). [Hint: Your equation should look like the preceding one with some additional terms]
. Consider the case where the masses are equal. a. Establish expressions for the initial and final accelerations b. Establish expressions for the initial and terminal velocities c. Create qualitative sketches of the position, velocity, and acceleration versus time.
ANALYSIS AND CALCULATIONS 1. Compare the observed initial and final accelerations to the results given by the corresponding expressions from the theory section.. Compare and contrast the qualitative sketches created for the kinematic variables in the theory section with the data collected during the experiment.
3. Drag Coefficient a. Use the observations and the terminal velocity expression derived in the theory section to estimate the drag coefficient for the spherical object. b. The figure on the next page illustrates how dramatically the type of flow around the sphere can effect the drag coefficient. Very smooth flow (laminar flow) results from small velocities and/or viscous fluids. Very turbulent flow occurs for large velocities and/or non-viscous fluids. The dimensionless parameter that determines the type of flow is called the Reynolds Number Where: ρ = fluid density (Kg/m 3 ) V = speed of sphere (m/sec) D = diameter of sphere (m) μ = fluid viscosity (kg/m sec) i. Use the data collected for this experiment to calculate the Reynolds Number for the sphere as it moves through the water at its maximum velocity ii. Use the graph on the following page to estimate the associated drag coefficient. c. Compare the drag coefficients you found in a. and b.
SPHERE CYLINDER