Teacher Notes


Teacher Notes
Publication No. 12650
Projectile Motion with an Inclined PlaneAdvanced Student Laboratory KitMaterials Included In Kit
Balls, steel, ¾" diameter, 8
Metal sheets, 4" x 4", 8 Washers, 8 Additional Materials Required
Clamp holder
Fishing line, or thin string, 1.5–2 m* Inclined plane, wood* Meter stick Pencil or chalk Printer paper, white, 3–4 sheets Protractor Scissors Support rods* Support stand Support stand clamp Textbooks, 3–4 (optional) Transparent tape *Inclined Plane—Classroom Set (Catalog No. AP6685) includes 8 wood inclined planes, 8 support rods and thin string. Safety PrecautionsRemind students to quickly retrieve the ball once it hits the floor. Wear safety glasses. Please follow all laboratory safety guidelines. DisposalThe materials should be saved and stored for future use. Lab Hints
Teacher Tips
Sample DataHeight of the tabletop from the floor (H): ___83.3 cm___ {12650_Data_Table_1}
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Student PagesProjectile Motion with an Inclined PlaneIntroductionUse knowledge of projectile motion and the law of conservation of energy to determine the launch speed of a steel ball after rolling down an inclined plane. Concepts
BackgroundWhat Galileo (1564–1642) proposed and Newton (1643–1727) essentially proved is that ALL objects fall toward the Earth at the same increasing rate (if in a vacuum). That is, all objects will accelerate toward the Earth equally, regardless of their mass. At the surface of the Earth, the acceleration of all objects toward the center of the Earth is measured to be (on average) 9.81 m/s^{2} (32 ft/s^{2}). {12650_Background_Equation_1}
H = height (sometimes referred to as Δy) {12650_Background_Equation_2}
The time calculated in Equation 2 is the total time it takes for the ball to fall from the edge of the tabletop to the floor. Since the horizontal speed is constant, the distance the ball travels horizontally can be determined by multiplying the horizontal speed by the total flight time of the ball (Equations 3 and 4).
{12650_Background_Equation_3}
D = horizontal distance (sometimes referred to as Δx) {12650_Background_Equation_4}
The initial horizontal speed can then be evaluated by rearranging Equation 4 to solve for v_{h}.
{12650_Background_Equation_5}
Equation 5 will be used to calculate the actual speed of the ball as it leaves the tabletop.How will the actual speed compare to the theoretical speed? To determine the theoretical speed at which the ball should travel as it leaves the edge of the tabletop after rolling down the inclined plane, the potential and kinetic energy of the ball on the inclined plane need to be evaluated. In order to raise the ball to the release point on the inclined plane, one must exert energy. Work, another term for energy, is being performed on the ball in order to raise it. Work is defined as a force used through a distance. It requires work to move any object. If a force is used on an object that doesn't move, such as pushing on a brick wall, then no work is being done. The energy used to raise the ball to a higher position is "stored" in the ball—the ball is said to have potential energy (PE). This potential energy may be used at a later time to do work. The potential energy of the ball is related to its height and weight. The potential energy 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 bottom of the inclined plane (Equation 6). {12650_Background_Equation_6}
As the ball begins to roll down the inclined plane, its potential energy is converted into kinetic energy, or energy of motion. The ball experiences two different types of motion—it is traveling in a straight path down the inclined plane and it is rotating. Therefore it has both linear kinetic energy and rotational kinetic energy. Linear kinetic energy (KE_{l}) is related to the mass (m) and linear speed (v) of the object (Equation 7). Rotational kinetic energy (KE_{r}) is related to the moment of inertia (I) of the ball and the rotational speed (ω; the Greek letter “omega”). See Equation 8. (Notice the similarity between Equation 7 and Equation 8.) The moment of inertia is the “resistance” an object has to being rotated. It 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 the center of the ball. The total kinetic energy (KE_{T}) of a rolling ball is therefore equal to the linear kinetic energy plus the rotational kinetic energy (Equation 9).
{12650_Background_Equation_7}
{12650_Background_Equation_8}
{12650_Background_Equation_9}
The law of conservation of energy states that energy cannot 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 inclined plane (neglecting frictional forces). This is represented by Equation 10.
{12650_Background_Equation_10}
Substituting Equation 6 and Equation 9 into Equation 10 yields
{12650_Background_Equation_11}
Equation 11 can be used to determine the theoretical speed of the rolling ball when it reaches the bottom of the inclined plane.The moment of inertia of a solid sphere rotating about its center is equal to 2/5mR^{2}, where R is equal to the radius of the sphere. Rotational speed, ω, is related to linear speed of the ball, v, and the radius, R, of the object: ω = v/R. Substituting these values into Equation 11 yields {12650_Background_Equation_12}
Rearranging Equation 12 to solve for v^{2}
{12650_Background_Equation_13}
Notice that Equation 13 shows that the speed of the ball at the bottom of the inclined plane is independent of the size or the mass of the solid ball. This means that any size solid ball will have the same speed at the bottom of the inclined plane so long as it is released from the same height. Equation 13 will be used to calculate the theoretical speed of the ball at the bottom of the inclined plane.
Experiment OverviewRoll a ball down an inclined plane off the end of a tabletop and measure the distance the ball falls. Use these measurements to calculate the initial speed of the ball, and compare this value to the theoretical speed calculated using the conservation of energy principle. Materials
Ball, steel, ¾" diameter
Clamp holder Inclined plane, wood Metal sheet, 4" x 4" Meter stick Pencil or chalk Printer paper, white, 3–4 sheets Protractor Scissors Support rod Support stand Support stand clamp Textbooks, 3–4 (optional) Transparent tape Washer and fishing line (about 1.5–2 m) (for plumb bob) ProcedurePreparation
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