1. Theoretical background In this lab session, we will study how the friction affects Uniformly Accelerated Motion (ASU) in an inclined plane (Figure 1). The aim of the session will be to determine the acceleration of the moving object by measuring the velocity at different locations of the inclined plane. In addition, another objective will be to evaluate how the acceleration depends on the gravity and friction force. Fig. 1: Picture of the experimental setup used in this laboratory session. If we consider the motion of a solid block of mass m in the inclined placed with angle (in the friction-free platform or air rail), see the diagram in Figure 2. The block is released in the reference position that we will define as d = 0 with no velocity. Due to the action of gravity the solid block will start moving along the inclined plane. We will determine the velocity of the solid block v at different locations d by using a photodiode provided. v d m Fig. 2: Schematic diagram of the experiment that shows some of the key parameters. The solid block will experience a Uniformly Accelerated Motion with acceleration a. Following 1
the kinetics laws, as described in the theory sessions, with an initial velocity equals zero we can express the velocity as a function of the acceleration as v 2 2ad (1) We will define the velocity of the block v when the block has moved a distance d along the plane in fig. 2. Considering this, the equation allows us to calculate the acceleration if we know the velocity at different distances from the initial point. We can also demonstrate that a g sin cos (2) where g is the gravity acceleration and represents the dynamic friction coefficient. Our objective is to measure v for different values of d and calculate the acceleration. Errors in the calculation of a are minimized by using Least Squares Method to adjust the measured data to the observed linear tendency. With the value of a and with the elevation angle, which we can measure, we will calculate the value of. We will observe that the acceleration that the solid block experience is independent of the mass of the block. 2. Experimental procedure The friction-free platform is in contact with the bench by two circular holders. Underneath one of this holder we will place a small circular weight of known length which will incline the friction-free platform. The photoelectric cell will be mounted to measure the speed of the solid block (skate) (Fig. 3). We will hold the skate at the chosen reference point, d = 0, where we will release the skate in such a way that the initial velocity is v0 = 0. Fig. 3. Photoelectric cell setup It is convenient to work with maximum air pressure in the friction-free platform, so we minimize the effect of friction. 2
1. With no extra weight on the skate we will put in place the rectangular flag (Fig. 3). bandera Fig. 4. The skate represents the solid block and will be equipped with a flag 2. We will fix the photoelectric cell at the distance d = 10 cm from the origin. CCR t m d Fig. 4. Montaje de la medida 3. We will hold the skate with the hand at the origin and we will release it with no initial impulse. We will read the time t on the photoelectric cell and we will write down the value in a table in your laboratory notebook. We will repeat the measurement three times to have statistical data on the time that takes the flag to cross the photoelectric cell (t ) At this stage it is not necessary to calculate the error, it is sufficient to take an adequate number of significant figures. Note that the recorded time is the time t that takes the flag to pass through the photoelectric cell, from one side of the flag that crosses the beam until the other side stops crossing the beam, as we know the width of the flag we can calculate the speed of the skate v = d flag t (3) 3
Where d flag is the width of the flag, we will take a value of d flag = 2.50 cm. We will write down in the notebook the velocity of the skate. 4. In order to use equation 1 to calculate the acceleration we need to measure the velocity at different distances from the origin. Then, we will move the photoelectric cell at intervals of 10 cm from the origin, measure the velocity of the skate three times. Measure the velocity at 6 different distances from the origin. Note that each distance d refers to the distance that the skate will travel from the moment we release it to the friction-less platform to the moment the middle of the flag reaches the photoelectric cell beam. In practical terms the middle of the skate (and middle of the flag) will be released at the reference (d=0) so the measured distance corresponds to the distance from reference to the middle of the flag. 5. Realizamos el experimento anterior para diferentes valores de la posición de la célula d. 6. We will prepare a table in the lab book that contains all the results: the three values of t; t and v. If we pay attention to equation (1) we will realize that it is the equation of a straight line in ther plot v 2 (t). We can define x d (4) And y v 2 (5) Equation (1) can be reformulated in the form y b0 b1 x (6) And this is the equation of a straigth line with the intercept b0 = 0 and a slope b1 = 2a. 7. Calculate the values of x and place values of x and y as new entries of the table. Represent y(x) graphically. From equation (5) y (6) we can observe that by adjusting the values of (y,x) By Least Squares Method we can obtain a from the slope b1 with its error. 8. By Least Squares Method determine the value of the acceleration a and its error. 9. The friction coefficient is calculated from equation (2). First, it is necessary to calculate by 4
using trigonometry and a ruler provided to measure the relevant sides of a triangle. For example, sin h/ L, with L the distance between the two circular holders of the frictionless platform, and h the height of the weight used to produce the inclined plane. 3. Data recording and presentation of results In the notebook it is necessary to describe the procedure followed to obtain the data, the table with the results, the graph x-y with the linear regression, and the determined value for a and together with their errors. Indicate the most relevant calculations that justify the results. 4. Additional questions 4.1. Discuss the advantages and drawbacks in the proposed method to determine the dynamic fiction coefficient. 4.2. Considering the calculate error do you think that the calculated coefficient friction is accurate? 4.3. Discuss if the assumptions used in this experiments are reasonably valid. For example, the assumption that the velocity is constant whilst the flag crosses the photodiode. 4.4. How would the experiment be affected if the length of the flag was 10 cm instead of 2.5 cm? 4.5. Propose an alternative method to calculate the dynamic friction coefficient. 5