Teacher Notes



Teacher Notes
Publication No. 12520
PSworks™ Marble RampStudent Laboratory KitMaterials Included In Kit
Ballbearings, steel, ¾" dia., 2 (marble)
Knob with threaded stud PSworks™ Marble Ramp Additional Materials Required
Fishing line or string, 1–2 m (for plumb bob)
Graph paper Meter stick Pencil or chalk Printer paper, white, 3–4 sheets Protractor PSworks™ Photogate Timer PSworks Support Stand Ruler (calipers, optional) Scissors Tape, masking Washer (for plumb bob) Safety PrecautionsThe materials in this lab are considered safe. Please follow all other laboratory safety guidelines. DisposalThe materials should be saved and stored for future use. Lab Hints
Teacher Tips
Correlation to Next Generation Science Standards (NGSS)^{†}Science & Engineering PracticesAnalyzing and interpreting dataUsing mathematics and computational thinking Developing and using models Planning and carrying out investigations Constructing explanations and designing solutions Obtaining, evaluation, and communicating information Disciplinary Core IdeasMSPS3.A: Definitions of EnergyMSPS3.B: Conservation of Energy and Energy Transfer MSPS3.C: Relationship between Energy and Forces MSPS2.A: Forces and Motion MSPS2.B: Types of Interactions HSPS3.A: Definitions of Energy HSPS3.B: Conservation of Energy and Energy Transfer HSPS3.C: Relationship between Energy and Forces HSPS2.A: Forces and Motion HSPS2.B: Types of Interactions Crosscutting ConceptsEnergy and matterCause and effect Systems and system models Patterns Performance ExpectationsHSPS21: Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. Sample DataPart 1. Forces and Gravity {12520_Data_Table_1}
Part 2. Projectile Motion Release Point Height Release Point 1: ___49.1 cm___ 2: ___43.7 cm___ 3: ___33.2 cm___ Height of the bottom of ramp: ___5.2 cm___ Height of the edge of the Marble Ramp to the floor: ___95.8 cm___ Marble Ramp Angle: ___35°___ {12520_Data_Table_2}
Answers to QuestionsPart 1. Forces and Gravity
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Student PagesPSworks™ Marble RampIntroductionExperiment with inclined planes and projectiles to determine how the acceleration due to gravity affects falling objects. Concepts
BackgroundGravity and Forces {12520_Background_Equation_1}
{12520_Background_Equation_2}
F_{x} and F_{y} are the horizontal and vertical components of the force, F, respectively. The Greek letter theta (θ) represents the angle between the force and the xcoordinate (see Figure 1).
{12520_Background_Figure_1}
Equations 1 and 2 can be used for any vector quantity, including acceleration and velocity. A vector quantity is a value that has both a magnitude and a direction associated with it. For example, a ball traveling down an inclined plane will accelerate due to the force of gravity, and the forces are represented in Figure 2. Note: As a convention, the yaxis is positioned parallel to the direction of the ball traveling down the inclined plane. Orientating the axis in this manner allows one to use Equations 1 and 2. (Recall that F = ma.) Part 1 of this experiment will use an inclined plane (ramp) and a ball (marble) to determine the acceleration due to gravity.
{12520_Background_Figure_2}
Projectile Motion When a ball is in “free fall,” the only force acting on it is the downward pull due to gravity. Therefore, if the “free falling” ball has any initial horizontal speed, the horizontal speed will remain constant throughout the free fall. The vertical speed of the ball, however, will change the instant it begins to fall. The constant acceleration due to gravity will cause the ball’s vertical speed to change at a constantly increasing rate. In Part 2 of this experiment, a ball will travel down an inclined plane and launch horizontally off the end of a table. The ball’s speed at the bottom of the inclined plane, just as it leaves the edge, will be calculated by measuring its flight distance. The first value that is needed is the time the ball is in free fall. Because the acceleration due to gravity is the same for all objects, the time it takes for any object, initially at rest, to fall a specific distance will be the same. Therefore, the distance any falling object travels in a given amount of time can be determined using Equation 3. {12520_Background_Equation_3}
H = falling height {12520_Background_Equation_4}
The time calculated in Equation 4 is the “free fall” time of the ball to fall from the bottom edge of the inclined plane to the floor (H). Since the ball’s horizontal speed will be constant, the distance the ball travels horizontally can be determined by multiplying the horizontal speed by the flight time of the ball (Equations 5 and 6).
{12520_Background_Equation_5}
D = horizontal distance {12520_Background_Equation_6}
And finally, the horizontal speed of the ball in flight can be evaluated by rearranging Equation 6 to solve for v_{h}. Equation 7 will be used to calculate the actual speed of the ball as it leaves the edge of the inclined plane.
{12520_Background_Equation_7}
Conservation of Energy How does the experimental speed of the ball at the bottom of the inclined plane compare to the theoretical speed? The theoretical speed at which the ball should travel as it leaves the edge of the inclined plane can be calculated by evaluating the potential and kinetic energy changes of the ball as it travels down the inclined plane. Work is the act of using a force to move an object through a distance. In order to raise the ball to the “release point” on the inclined plane, one must exert energy (work) to lift it. The energy expended to raise the ball to a higher position is “stored” in the ball—the ball now has potential energy (PE). The potential energy of the ball is related to its height and weight, and is equal to the mass (m) of the ball multiplied by the acceleration due to gravity (g) multiplied by the relative height (h) of the ball above the ground, or in the case of this experiment the bottom edge of the inclined plane (Equation 8). {12520_Background_Equation_8}
As the ball rolls down the inclined plane, its potential energy is converted into kinetic energy, or energy of motion. This is due to the conservation of energy principle. When the ball rolls it has two different types of kinetic energy—linear and rotational. Linear kinetic energy (KE_{l}) is related to the mass (m) and linear speed (v) of the object (Equation 9). Rotational kinetic energy (KE_{r}) is related to the moment of inertia (I) of the object and the rotational speed (ω; the Greek letter omega). See Equation 10. The moment of inertia of an object is its “resistance” to being rotated. This “resistance” is based on the mass of the object and the spatial distribution of the mass about the point of rotation. The point of rotation for the ball is its center. The total kinetic energy (KE_{T}) of a rolling ball is therefore equal to the linear kinetic energy plus the rotational kinetic energy (Equation 11). {12520_Background_Equation_9}
{12520_Background_Equation_10}
{12520_Background_Equation_11}
The conservation of energy principle states that energy can not be created or destroyed—energy can only be converted from one form to another. Therefore, the initial potential energy the ball has at the release point will be completely converted into kinetic energy at the bottom of the ramp (neglecting frictional forces). This is represented by Equations 12 and 13. Equation 13 will be used to determine the theoretical speed of the rolling ball when it reaches the bottom edge of the inclined plane just as it launches into free fall.
{12520_Background_Equation_12}
{12520_Background_Equation_13}
The moment of inertia of a solid ball rotating about its center is equal to (2/5)mR^{2}, where R is equal to the radius of the sphere. Rotational speed, ω, is related to the linear speed of the ball, v, and the radius, R, of the object: ω = v/R. Substituting these values into Equation 13 gives Equation 14.
{12520_Background_Equation_14}
Reducing and rearranging Equation 14 to solve for v^{2} generates
{12520_Background_Equation_15}
Notice that Equation 15 shows that the speed of the ball at the bottom of the inclined plane is independent of the size or the mass of the ball. Any size solid ball will have the same speed at the bottom of the inclined plane as long as it released from the same height. Equation 15 will be used to calculate the theoretical speed of the ball at the bottom edge of the Marble Ramp just before it begins to free fall to the floor.
Experiment OverviewIn Part 1 of this experiment, use the Marble Ramp and a Photogate Timer to determine the acceleration due to gravity. In Part 2, use the Marble Ramp to launch a marble (ball) horizontally off the end and calculate the speed of the marble using the distance it travels and height it falls. Then, compare this value to the theoretical value using the conservation of energy principle. Materials
Ball, steel, ¾" dia. (marble)
Fishing line, 1–2 m (for plumb bob) Graph paper Knob with threaded stud Meter stick Pencil or chalk Printer paper, white, 3–4 sheets Protractor PSworks™ Marble Ramp PSworks Photogate PSworks Support Stand Ruler Scissors Tape, masking Timer Washer (for plumb bob) Safety PrecautionsThe materials in this lab are considered safe. Please follow all normal laboratory safety guidelines. ProcedurePart 1. Forces and Gravity
Setup
When performing this experiment, one lab partner will place the marble at the specific release point on the Marble Ramp as noted in step 7, and then release it. A second lab partner needs to be in a position to catch the marble after it launches off the end of the ramp and hits the floor.
Student Worksheet PDF 