Newton's 2nd Law Of Motiom

1 I. OBJECTIVES



To know how total mass affect the acceleration of an object with constant force.



To know how force affect the acceleration of an object with constant total mass.



To prove Newton’s Second Law of Motion

II. SET-UP

2 III. THEORY Newton's Second Law Newton's Second Law as stated below applies to a wide range of physical phenomena, but it is not a fundamental principle like the Conservation Laws. It is applicable only if the force is the net external force. It does not apply directly to situations where the mass is changing, either from loss or gain of material, or because the object is traveling close to the speed of light where relativistic effects must be included. It does not apply directly on the very small scale of the atom where quantum mechanics must be used. Data can be entered into any of the boxes below. Specifying any two of the quantities determines the third. After you have entered values for two, click on the text representing to third to calculate its value.

Newton's Second Law Illustration Newton's 2nd Law enables us to compare the results of the same force exerted on objects of different mass.

3

Newton's second law: law of acceleration The rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction. In Motte's 1729 translation (from Newton's Latin), the Second Law of Motion reads: LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. — If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined,

4 when they are oblique, so as to produce a new motion compounded from the determination of both. Using modern symbolic notation, Newton's second law can be written as a vector differential equation: where: is the net force vector is mass is the velocity vector is time. The product of the mass and velocity is the momentum of the object (which Newton himself called "quantity of motion"). The use of algebraic expressions became popular during the 18th Century, after Newton's death, while vector notation dates to the late 19th Century. The Principia expresses mathematical theorems in words and consistently uses geometrical rather than algebraic proofs. If the mass of the object in question is constant, this differential equation can be rewritten as: where: is the acceleration.

5 A verbal equivalent of this is "the acceleration of an object is proportional to the force applied, and inversely proportional to the mass of the object". If m is dependent on velocity (and thus indirectly upon time) as we now know it is (for high velocities—see special relativity), then m has to be included in the derivative, as above. Taking Special Relativity into consideration, the equation will become where: m0 is the rest mass or invariant mass. Note that force will depend on speed of the moving body, acceleration and its rest mass. However, when the speed of the moving body is much lower than the speed of light, the equation that was shown above will be reduced to our familiar Contrary to what is sometime claimed in elementary texts, mass must always be taken as constant in classical mechanics. So-called variable mass systems like a rocket can not be directly treated by making mass a function of time in the second law. The reasoning, given in An Introduction to Mechanics by Kleppner and Kolenkow and other modern texts, is excerpted here: Newton's second law applies fundamentally to particles. In classical mechanics, particles by definition have constant mass. In case of well-defined systems of particles, Newton's law can be extended by integrating over all the particles in the system. In this case we have to refer all vectors to the center of

6 mass. Applying the second law to extended objects implicitly assumes the object to be a well-defined collection of particles. However, 'variable mass' systems like a rocket or a leaking bucket do not consist of a set number of particles. They are not well-defined systems. Therefore Newton's second law can not be applied to them directly. The naive application of F = dp/dt will usually result in wrong answers in such cases. However, applying the conservation of momentum to a complete system (such as rocket+fuel, or bucket+leaked water) will give unambiguously correct answers. Newton's Second Law Newton's first law of motion predicts the behavior of objects for which all existing forces are balanced. The first law - sometimes referred to as the law of inertia - states that if the forces acting upon an object are balanced, then the acceleration of that object will be 0 m/s/s. Objects at equilibrium (the condition in which all forces balance) will not accelerate. According to Newton, an object will only accelerate if there is a net or unbalanced force acting upon it. The presence of an unbalanced force will accelerate an object - changing either its speed, its direction, or both its speed and direction.

7

Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is decreased.

Newton's second law of motion can be formally stated as follows:

8 The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. This verbal statement can be expressed in equation form as follows: a = Fnet / m The above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration. Fnet = m * a IV. PROCEDURE

First, we fixed a good running pulley about 1.5 meters from the floor. Passed a piece of string over the pulley and at each end, tight snugly a 100-gram mass. An extra 10 grams to the left was added so that the actual moving net force is due to weight of the additional 10 grams. The mass was held to prevent its descent. Then, we found the time t (in second) for the heavier mass to descend from the starting point to the floor. Stopwatch started simultaneously with the release of the heavier mass. Be sure that the time recorded is only up to the time that the mass touches the floor, not after it has fallen to the floor. We measured the distance S through which the left mass had descended and calculated the acceleration from the formula: S= ½ a t2 or a= 2S/ t2. Then, we considered the heavier body (going down) as m1 and the lighter body (going up)

9 as m2 and computed the acceleration from the formula: a= [(m 1-m2)g]/ m1+m2. The procedure 2 was repeated by using 20-gram weight as the actual moving force and we also repeated steps 3 to 5. For the third trial, we combined the two loads used in steps 2 and 6 as the actual moving force and repeated steps 3 to 5. All data and results were entered in the table.

V. DATA/OBSERVATIONS Table A: Trial

Total Time (dots)

Total Distance (cm)

Time Interval (dots)

Trial 1

9 18 27 36 45 9 18 27 36 45 9 18 27 36 45 9 18 27 36 45

1 2.9 63.4 10.9 16.2 0.9 2.5 4.6 7.6 11.3 0.85 2.35 4.4 7.1 10.3 0.8 2.3 4.35 6.9 9.9

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

Trial 2

Trial 3

Trial 4

Distance Traveled during each Time Interval (cm) 1 1.9 3.5 4.5 5.3 0.9 1.6 2.1 3 3.7 0.85 1.5 2.05 2.7 3.2 0.8 1.5 2.05 2.55 3.0

Average Average Acceleration 2 Acceleration Speed (cm/dot ) (cm/dot2) during each Time Interval (cm/dot) 0.11 0.011 0.21 0.02 0.013 0.39 0.012 0.5 0.59 0.01 0.1 0.009 0.18 0.006 0.009 0.23 0.01 0.33 0.41 0.009 0.094 0.008 0.17 0.007 0.008 0.23 0.008 0.3 0.36 0.007 0.089 0.009 0.17 0.007 0.007 0.23 0.006 0.28 0.33 0.006

10

Table B: Trial

Total Time (dots)

Total Distance (cm)

Time Interval (dots)

Trial 1

9 18 27 36 45 9 18 27 36 45 9 18 27 36 45 9 18 27 36 45

0.8 2.3 4.35 6.9 9.9 1.1 5.44 12.21 20.60 31.40 2.9 10.5 22.3 38.3 58.3 3.65 13.95 30.45 52.85 80.95

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

Trial 2

Trial 3

Trial 4

Given: (washers) 13 & 22 = 67g 21 & 25 = 65g 7 & 24 = 67g 11 & 12 = 63g Cart = 469g

Distance Traveled during each Time Interval (cm) 0.8 1.5 2.05 2.55 3.0 1.1 4.34 6.77 8.39 10.8 2.9 7.6 11.8 16 20 3.65 10.3 16.5 22.4 28.1

Average Average Acceleration Acceleration Speed (cm/dot2) (cm/dot2) during each Time Interval (cm/dot) 0.089 0.009 0.17 0.007 0.007 0.23 0.006 0.28 0.33 0.006 0.122 0.04 0.482 0.03 0.03 0.752 0.02 0.932 1.2 0.03 0.32 0.058 0.84 0.052 0.053 1.31 0.052 1.78 2.22 0.049 0.406 0.082 1.144 0.077 0.076 1.833 0.073 2.489 3.122 0.070

11 OBSERVATIONS: In the first set-up: 

As the total mass increases, the total distance decreases.



As the total mass increases, the velocity decreases.



As the total mass increases, the average acceleration decreases.



The distance between the dots decreases as the mass of the cart is moving slower when it has greater mass.

In the second set-up 

As the force applied increases, the total distance also increases.



As the force applied increases, the velocity also increases.



As the force applied increases, the average acceleration also increases.



The distance between dots increases as the forces pulling it increases showing that the cart is moving faster when greater force is applied to make it move.

VI. COMPUTATIONS/GRAPH Table A Trial

Force

Mass

T1

11&12

cart

T2

11&12

13&22 + cart

T3

11&12

13&22 + 25&21 + cart

T4

11&12

13&22 + 25&21 + 7&24 + cart

12

Table B

Trial

Force

Mass

T1

11&12

13&22 + 25&21 + 7&24 + cart

T2

11&12 + 7&24

13&22 + 25&21 + cart

T3

11&12 + 7&24 + 25&21

13&22 + cart

T4

11&12 + 7&24 + 25&21 + 13&22 + cart

cart

Sample Computation:

Average Speed during each Time Interval (cm/dot): For table A:

Trial 1 1cm / 9 dots = 0.11 cm/dot Trial 2 0.9cm / 9 dots = 0.1 cm/dot Trial 3 0.85cm/ 9 dots = 0.94 cm/dot Trial 4 0.8cm / 9 dots = 0.089 cm/ dot

For table B:

Trial 1 0.8cm / 9 dots = 0.089 cm/dot Trial 2 1.1cm / 9 dots = 0.122 cm/dot Trial 3 2.9cm / 9 dots = 0.32 cm/dot Trial 4 3.65cm / 9 dots = 0.406 cm/ dot

13

14 Acceleration (cm/dot2)

For Table 1

T1

(1.9cm -1 cm) / (9 dots) 2 = 0.011 cm/dot2

T2

(1.6cm -0.9 cm) / (9 dots) 2 = 0.009 cm/dot2

T3

(1.9cm -0.85 cm) / (9 dots) 2 = 0.008 cm/dot2

T4

(1.9cm -0.8 cm) / (9 dots) 2 = 0.009 cm/dot2

For Table 2

T1

(1.5cm -0.8 cm) / (9 dots) 2= 0.009 cm/dot2

T2

(4.34cm -1.1 cm) / (9 dots) 2 = 0.04 cm/dot2

T3

(7.6cm -2.9 cm) / (9 dots) 2 = 0.058 cm/dot2

T4

(10.3cm -3.65 cm) / (9 dots) 2= 0.082 cm/dot2

Average Acceleration

For Table 1 = (0.11cm/dot2 + 0.02cm/dot2 + 0.012 cm/dot2 + 0.01 cm/dot2) / 4 = 0.013 cm/ dot2 For Table 2 = (0.009 cm/dot2 + 0.007 cm/dot2 + 0.006 cm/dot2 + 0.006 cm/ dot2) / 4 = 0.007 cm/dot2

15 VII. ANALYSIS/ DISCUSSION/ QUESTION

As shown by the graph in first set-up, it can be analyzed that as the mass increases having constant force the acceleration of the body decreases. This shows that as the total mass increases the body appears to move slowly. While the graph of the second set-up shows that as the force increases with constant total mass, the acceleration of the body also increases. This mean that body appears to move faster as force pulling it increases.

VIII. CONCLUSION

Upon performing this experiment, we therefore conclude that an increase in total mass having constant force decreases the acceleration of a body and vice versa; and an increase in force with constant total mass, increases the acceleration of a body and vice versa. With these conclusions, we were able to prove that a= F/m; this means that the acceleration of a body is directly proportional to the force applied but inversely proportional to its total mass.

16 IX. APPLICATION The V-2 Rocket The V-2 military rocket, used by Germany in 1945, weighed about 12 tons (12,000 kg) loaded with fuel and 3 tons (3,000 kg) empty. Its rocket engine created a thrust of 240,000 N (newtons). Approximating g as 10m/s2, what was the acceleration of the V-2 (1) at launch (2) at burn-out, just before it ran out of fuel? Solution

Let the upwards direction be positive, the downwards direction

negative: using this convention, we can work with numbers rather than vectors. At launch, two forces act on the rocket: a thrust of +240,000 N, and the weight of the loaded rocket, mg = –120,000 N (if the thrust were less than 120,000 N, the rocket would never lift off!). The total upwards force is therefore F = + 240,000 N – 120,000 N = +120,000 N,

and the initial acceleration, by Newton's 2nd law, is a = F/m = +120,000 N/12,000 kg = 10 m/s2 = 1 g

The rocket thus starts rising with the same acceleration as a stone starts falling. As the fuel is used up, the mass m decreases but the force does not, so we expect a to grow larger. At burn-out, mg = –30,000 N and we have

17 F = + 240,000 N – 30,000 N = +210,000 N,

giving a = F/m = +210,000 N/3,000 kg = 70 m/s2 = 7 g

The fact that acceleration increases as fuel is burned up is particularly important in manned spaceflight, when the "payload" includes living astronauts. The body of an astronaut given an acceleration of 7 g will experience a force up to 8 times its weight (gravity still contributes!), creating excesive stress (3-4 g is probably the limit without special suits). It is hard to control the thrust of a rocket, but a rocket with several stages can drop the first stage before a gets too big, and continue with a smaller engine. Or else, as with the space shuttle and the original Atlas rocket, some rocket engines are shut off or dropped, while others continue operating.

18 X. REFLECTION

Our group enjoyed doing this experiment. This experiment was a successful product of the collective efforts of the Velocity Group. Each of us cooperates and shares our knowledge to each member of the group. Because of this we were able to accomplish our tasks with our professor’s guidance. All of us were excited to perform this experiment. Before the experiment started, we were all excited and at the same time, nervous because we had no idea how the outcome will turn out. Even though we felt nervous, it doesn’t affect our performances in this activity. When we finished performing this experiment, all of us were relieved. And somehow, we were satisfied with the results because we know that we did our best. This experiment didn’t give us a hard time because we totally enjoyed doing it. We also have a little background in this topic so it is not hard for us to do this experiment. We already tackle this topic when we are in high school. This experiment in the laboratory helped us a lot to build a wonderful relationship with each other. Aside from having a good relationship, we also learned the sense of fairness and responsibility. By distributing the tasks equally to each member of the group, we were able to accomplish this experiment.