EXPERIMENT 5 NEWTON S SECOND LAW WITH A CONSTANT MASS Objectives 1. To find the acceleration of a cart using the graph of its velocity versus time 2. To establish a mathematical relation between the acceleration of an object and the applied force when the mass of the object is held constant 3. To calculate the acceleration of an object given the net force and mass, and then compare it to the experimental value, and 4. To verify the relation between the acceleration, mass and net force on an object subjected to linear motion. Theory An important equation in physics is the mathematical form of Newton s second law, F = ma Where F is the net external applied force on an object of mass m and a is the resulting acceleration. Note that F and a are vectors. This law establishes the fact that only an external unbalanced force will cause an object to move and accelerate. The purpose of this experiment is to find out what happens to the acceleration of an object of mass m if an external force is applied. We will use a motion sensor to study the object motion while an external force is acting it upon. We will determine what happens to the acceleration of the object when the net force is changed while keeping the mass of the object constant. We shall keep in mind that the acceleration of the object is defined as the change of velocity with time i. e. it is the derivative of its velocity with respect to time. Δv dv a = lim = t 0 Δt dt What happens to an object when a net external force is applied? How does the motion of an object change when the magnitude of this force is changed? Newton s first law states that if the net force on an object is zero, then the object is at rest or moves with a constant speed with no change in its direction. The second law states that when a net force is not zero then the object will accelerate. If the external force is increased the acceleration will increase (assuming constant mass). If the net force is doubled, the acceleration is doubled i. e. the net force is directly proportional to the acceleration. F a Oftentimes, several forces act on an object simultaneously. In such cases, the vectorial sum of all the forces is the relevant force. Also, Newton s second law states that the acceleration is inversely proportional to the mass. a 1/m Equipment Motion sensor 1 Balance 1 Pasco 750 Interface 1 Pulley with string 1 Set of masses 1 Cart track 1 Track 1 1
Procedure Figure 5-1. Leveled cart track. The cart must remain at rest at any point on the track 1. Level the cart track on a horizontal table by adjusting the levelling screws. Make sure the cart is stationary at any position on the track. See Figure 5-1 2. Note that the track is designed to accomodate the motion sensor at either end. Install a motion sensor at one end 3. Install the pulley at the other end and use a piece of wood to prevent the cart from hitting the pulley 4. Tie a string (about 1.2 m in length) to the cart and pass it over the pulley. See Figure 5-2 5. Measure the mass of the mass holder Experiment 1: 1. Place a 20 g mass on the mass holder. Write the total mass attached to the mass holder, in Table 1 of your lab report 2. Measure the mass of the cart 3. Place two masses of 20 g each on the cart 4. In your lab report, write the mass of the cart plus 40 g of the additional masses 5. Check that the setup of your experiment looks as that shown in the Figure 5-2 below Figure 5-2. Complete experimental setup for Newton s Second Law 6. Adjust the pulley so that the string is parallel to the track 7. Keep the cart a distance of at least 15 cm from the front of the sensor before releasing it 8. You will do three experiments using the above mentioned setup: i. In the first experiment you have 20 g on the mass holder and 40 g on the cart, ii. In the second experiment, transfer 20 g from the cart on to the mass holder. The total mass of the system is constant but the net force increases, iii. In the third experiment, transfer the other 20 g from the cart on to the mass holder. Once again the total mass of the system is constant but the net force increases Note: Remember to catch the cart before it hits the pulley or you may damage the cart and pulley 9. Press the Start button. Release the cart. Stop the cart before it hits the pulley and press the Stop button to stop the program. In the analysis section you will learn how to analyze the data Experiment 2. 10. For the second graph, remove 20 g from the cart and place it on to the mass holder. Write the total hanging mass in table 1 of your lab report Experiment 3. 11. For the third graph, remove the 20 g from the cart and place it on the mass holder. Write the total hanging mass in your lab report 2
Analysis Graphical analysis 1. Press Adjust Scale to make the graph appear more accurate if necessary. Use the cursor to select the linear portion of the curve 2. Select fit from the menu and select the linear fit option. The slope m of the selected linear portion of the graph is the acceleration 3. After calculating the slope of the graph in Figure 3 use the data menu to select the other graphs. The example in the plot below shows the linear fit and the fitting parameters 4. Make a note of the slope which is the acceleration for all the graphs 5. Follow the same procedure to calculate the acceleration for all of the graphs Figure 5-3 The slope of this line is the acceleration of the cart 3
Laboratory Report Of Experiment 5 Newton s Second Law with a constant mass Section Bench Students 1. 2. 3. 4. 1. Write the mass of the mass holder m h = kg 2. Add 20 g to the mass holder and write the total hanging mass and net external force in Table 1 for the three experiments. Also add 40 g to the cart 3. Fill out the remaining blanks in Table 1 with the data for experiments 2 and 3. Remember that you will transfer 20 g of mass from the cart to the mass holder each time Experiment 1 Mass of mass holder + added mass (kg) Table 1 Calculated net external force F = ma (N) Total mass in cart = mass of cart + added mass (kg) 2 3 4. Follow the procedure to graph the velocity of the cart versus time. Find the acceleration from the graph and enter its value in Table 2 below. Repeat the procedure for the other two experiments 4
Table 2 Graph Slope = Acceleration (m/s 2 ) 1 2 3 Data Analysis 1. What do you conclude from the slopes of the graphs when the net force on the cart increased but the total mass of the system remained constant? 2. Calculate the acceleration of the cart when the total mass of the system is held constant but the force changes. Use the equation below. The acceleration is the net force divided by the total mass a calculated = m ( m hanger + m + ( m added ) g + m total mass in cart hanger added ) Acceleration (Calculated) (m/s 2 ) Table 3 Acceleration (Experimental) (m/s 2 ) % Error 3. Draw a new graph of the net force (N) versus the acceleration (m/s 2 ) of the cart and calculate the % error of the total mass used (measured using a balance) and the total mass obtained from the slope of this graph. ΔF m = Δ a 5
Total mass obtained using a balance (kg) Table 4 Mass obtained from the slope of the graph (kg) % Error 4. What are your conclusions when the data are compared? Discussion 1. What variable (mass or force) did you keep constant during the experiment, and what variable did you change? Explain your answer 2. What variable changed as a result of keeping one constant? 6
Error analysis What are the sources of error in the experiment? Conclusions 1. What happens to the acceleration of the object (mass constant) if the net force is increased? 2. Does your experiment support this claim? Supplemental Problem A bicyclist has a mass of 87.1 kg, and his cycle has a mass of 13.1 kg. The bicyclist pedals his bicycle with a net horizontal force of 9.78 N. What is his acceleration? 7
Questions on Experiment 5 Newton s Second Law with a constant mass This questionnaire has some typical questions on experiment 5. All students who are taking the laboratory course of University Physics I must be able to correctly answer it before trying to make the experiment. 1. In a graph of velocity versus time, the slope gives: a. Instantaneous velocity b. Instantaneous acceleration c. Instantaneous position d. No known quantity e. Force 2. The velocity of an object is given by the following function v(t) = t 2 + 3t - 9 where t is the time in seconds. The acceleration of the object after 4 s is: a. 11 m/s 2 b. 19 m/s 2 c. Zero d. Cannot calculate, insufficient information e. 4.75 m/s 2 3. A larger acceleration requires: a. Larger mass b. Larger force c. Smaller mass d. Smaller force e. Not possible to tell 4. A 20 kg object accelerates at 5 m/s 2. The force required to maintain the acceleration constant is: a. 50 N b. 4 N c. 100 N d. There is no force e. 35 N 5. A net force of 2800 N is applied to a 1400 kg automobile at rest. The speed of the automobile after 10 s will be: a. 2 m/s b. 5 m/s c. 10 m/s d. 20 m/s e. 3 m/s 6. The following graph of velocity versus time represents that of an automobile that travels in a straight line. The distance traveled in the time interval between 2 s and 5 s is: Figure 4 A graph of velocity vs. time 8
a. 4 m b. 12 m c. 24 m d. 36 m e. 60 m 7. The following graph of velocity versus time represents that of an automobile that travels in a straight line. From this graph one can conclude that: Figure 5 A graph of velocity vs. time a. The automobile accelerates, then stops and then reverses direction b. The automobile accelerates at 6 m/s 2 during the first 2 s c. The automobile moves for a total of 12 s d. The automobile decelerates at 12 m/s 2 during the last 4 s e. The automobile returns to the starting point after 9 s 8. The mass of a salami of 3 lb is: a. 0.31 kg b. 1.36 kg c. 6.6 kg d. 29.4 kg e. 19.2 kg 9. One (1 kg)(1 m/1 s 2 ) is equivalent to: a. 1 m/s b. 1 Newton c. 1 Joule d. 1 vector e. 1 volt 10. The weight of an object is: a. The amount of mass that it contains b. The force with which it is attracted to the earth c. Basically the same as the mass but acts in the opposite direction d. A measure of its inertia e. Something that has the same value on any other planet 9