Measurements of us 100g mass 200g mass Run 1 30 25 Run 2 350 270 Run 3 33 30Lab S MASS ON INCLINED PLANE AND FRICTION OBJECTIVE: to measure the coefficient of static and kinetic friction between a block and an inclined plane and to examine the relationship between the plane's angle and its mechanical efficiency INTRODUCTION: Newton's second law of motion tells us that the net force on an object is equal to its mass times its acceleration (JF = ma); this equation can be applied to any spatial direction (x or y). The object is in equilibrium for a given direction if the sum of the forces in that direction is zero (EF = (). For an object that has mass, if the sum of the forces is zero, the acceleration of the object in that direction is necessarily zero. A non-zero acceleration can be accomplished in two ways: (1) the object could be at rest or (2) the object can be moving at a constant velocity. Consider the free-body diagram in Figure I that shows the forces acting on a block sitting on an inclined plane. In the y-direction (defined as perpendicular to the surface of the plane), there are two forces acting on the block; the sum of those forces must be equal to zero since the block is not moving in that direction. The force acting in the negative y direction is the component of the object's weight (W in figure I) that is in that direction, mg cos 0; g is the acceleration due to gravity, 9.8 m/s . It must be exactly balanced by the force acting upward on the block, called the normal force (or support force), N, that is defined as acting perpendicularly to the surface on which the block rests. Therefore, N = mg (cos 0) (eq. 1) There also is a component of the object's weight also acting in the x-direction, mg sin 0. If the object is at rest or it is moving at constant velocity down the plane (i.e. no acceleration), then there must be a force acting in the opposite direction which exactly balances the weight of the object in the x- direction such that the sum of the forces is zero, which we denote with Ff. This force is the force of friction, which resists the block's motion down the plane due to the interactions between molecules of the block and plane. Since the sum of the forces in the x-direction must equal zero, then the force due to friction, Fr, must be equal to the component of the block's weight that acts in the x-direction or Ff = mg (sin 0) (eq. 2) mg cos 200 W = W mg sin Figure 45The magnitude of the frictional force, Ff, on an object, can also be described by: Ff = UN (eq. 3) where u is the coefficient of friction. If the block is at rest, we say that the force of static friction, Fs is acting to counterbalance the weight component in the x-direction, and the coefficient of friction is that for the static case, us. If the block is in motion at a constant velocity, we say that the force of kinetic friction, Fk is acting on the block, and the coefficient of friction is that for the kinetic case, uk. Friction always opposes the direction of motion. Combining equations 1, 2, and 3 and solving for u, we have an equation for the angle where the force of friction is balanced with the weight component to give zero acceleration, which occurs at different angles for the two different cases of static and kinetic friction: H = Ff mgsine N = tane mgcose (eq. 4) EQUIPMENT: 100g and 200g masses Wood board and wood blocks Pulley with clamp attached String . . . . S hook Washers Protractor PROCEDURE: PART A: Measurement of us 1009 = 309 350 33 A1. Clamp the pulley to the front end of the board. Place it on a table and prop it up with a wood block or book. The pulley should be overhanging the table. The board should initially be at a small angle (around 10 degrees). Place the 100 g black mass on the board. Slowly increase the angle of the board by sliding your support. When the mass just barely begins to slide down the plane due to its own weight, record this angle 0s. Repeat two additional times, 200g : 25, 27: 30 recording all three angles. A2. Repeat the above step with the 200 g mass. 46A3. Put all data in a table. A4. You will not use a different method to measure us. Remove the support from under the board so it is laying flat on the table with the pulley overhanging it. Cut a piece of string about 12" long and tie the S hook to one end and a paper clip with one end opened up (as was done with the Atwood apparatus) to the other end. A5. Lay the string over the pulley with the S hook on the board. Attached the 100g mass to the S hook. Slowly add washers, one at a time, until the mass starts to move. Record the number of washers needed. & washers PART B: Application of Newton's Second Law with friction and gravitational forces In this part you will be creating a balanced force system with the board at an angle and validating the accuracy of us- B1. Place the support under the board. Place the 200 g mass on the board attached to the S hook and hang the 100 g mass from the paper clip. Hold onto the heavier mass, as the system will most likely not be balanced. B2. Adjust the angle of the board until the system is balanced. Record this angle. 36 . PART C: Measurement of coefficient of kinetic friction Ux C1. Place one of the masses on the board. Slowly increase the angle up to the point that when the block is pushed very lightly (just enough to overcome static friction), the mass will slide down the plane at constant speed. Be careful that the mass is sliding slowly with a constant speed and is not accelerating down the plane. Record this angle Ok. 40, 37, 42" C2. Repeat the previous step two additional times, recording all three angles. 47DATA ANALYSIS: 1. Calculate the average angles from Procedure steps A1 and A2. Calculate the coefficient of static friction. Also calculate us from each individual angle measurement. Is there any variation in us from the two masses? 2. You will now calculate us from procedure A4. Recall that the mass of one washer is 6.4 g a) Draw a free body diagram of the system at rest b) Write out the expression for the normal force on the mass on the board. c) Apply Newton's Law and solve for us d) Calculate us based on your measured data