Physics Lab 7 - The objective of this lab is to study Hooke’s law and determine the spring constant PDF

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The objective of this lab is to study Hooke’s law and determine the spring constant using the static and dynamic methods of measurements....

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PHYS 1433-General Physics I: Algebra

Hook’s Law and Spring Constant.

SECTION NUMBER: D721

DATE: 04/04/17

Objective:

The objective of this lab is to study Hooke’s law and determine the spring constant using the static and dynamic methods of measurements.

Theoretical Background: Springs are familiar objects that exhibits elastic behavior. For small deformations, once the deforming force is removed the spring will go back to its relaxed form. If we apply a force so that the spring is stretched, the spring exerts a force to restore itself to its original position. If the spring is compressed a restoring force is produced. In either case, the force of the spring acts to return the spring to its original position. The spring force is found to be proportional to the displacement x of the end of the spring from its problem when the spring is relaxed. It is represented by: F = -kx.

(10.1)

Hooke’s law is a mathematically accurate approximation of a certain force and should not be taken as a statement of a fundamental physical principle. For the helical spring, the spring constant k depends on the material and the geometry of the spring. We can find k experimentally by hanging various weights W at the end of a vertical spring and measuring the associated elongation x from its original position, as shown in Fig. 10.1. The spring force F and the hanging weight W balance each other. Therefore, the magnitude of the spring force is equal to the magnitude of the hanging weight: F = W = kx.

(10.2)

Thus, the magnitude of the hanging weight is directly proportional to the elongation x of the spring. A plot of the magnitude of the applied force W versus the elongation x gives a straight line that goes through the origin, as in Fig. 10.2. The slope of the straight line is equal to the spring constant k. In this way, we can experimentally determine the spring constant for any spring. This method of the determination of the spring constant is the static method. Let us consider a mass m hanging from a vertical spring of negligible mass. The weight of this mass causes the spring to stretch. In fact, the vertical spring is in equilibrium because it exerts an upward force equal to the weight of the mass. When the spring is in a new stretched position, a force pulls the mass to the equilibrium position. This force tends to restore the mass m to its original position and is defined by Hooke’s law (10.1), where the displacement x is defined as the distance the mass moves from its equilibrium position. Under the action of this force a mass m on a vertical spring oscillates about the equilibrium position. We can find the acceleration of the mass from Newton’s second law as

a=

F m

.

(10.3)

Combining equation (10.1) with equation (10.3), we have

a=-

kx m

or

a=-

k x. m

(10.4)

Thus, equation (10.4) shows that acceleration is directly proportional to the displacement x from the equilibrium position, and the acceleration of the system is not system is not constant but varies with x. Equation (10.4) is the defining equation for the simple harmonic motion. Simple harmonic motion is the motion whereby the acceleration is proportional to the displacement from the equilibrium position and is opposing directed. When a mass undergoes

simple harmonic motion, the restoring force acting on the vibrating mass is also proportional to its displacement and is oppositely directed.

ω

2

=

k m

ω

or

√

=

k m

.

(10.5)

Therefore, equation (10.4) becomes a=-

ω

2

x.

(10.6)

This type of equation, where the acceleration is directly proportional to x and is oppositely directed, has a simple solution for x, which is X = Acos( ω t + δ ) The argument of the cosine function and the constant

δ

ω t+ δ

(10.7)

is called the phase of simple harmonic motion

is called the phase constant. One complete motion is called a cycle. The

period T is the time it takes to complete one oscillation. During one complete oscillation, the phase increases by 2 π . At the end of the oscillation, the object again has the same position as it had at the start of the cycle since Cos( ω t + δ

+2 π

) = cos( ω t + δ

)

(10.8)

We can determine the period from the fact that the phase at the time t + T must be 2 phase at the time t: ω (t+T)+ δ

= ω t+ δ

So

T=

2π ω

.

+ 2 π or

ω T=2 π

(10.9)

π

plus the

An important relationship between the characteristics of the spring and the simple harmonic motion can be easily deduced from equations (10.5) and (10.10). Substituting

ω

from

equation (10.5) into equation (10.10), we obtain

√

T=2 π m k

(10.11)

As can be seen, the period is completely independent on the amplitude. You can start oscillations from any position and the period should always be the same. Equation (10.11) gives the period of simple harmonic motion In terms of the spring constant k and the moving mass m attached to the spring. Notice that for a particular value of m and k, the period of motion remains constant throughout the motion. It must be noted that equation (10.11) is only true when the mass of the spring m, is much less than the mass m, which is suspended from the spring. When the mass of the spring cannot be neglected, it can be shown that one third of the spring mass m, must be included in equation (10.11) along with m. As a result, the period of simple harmonic motion for a spring of finite mass m, becomes

T=2

π

√

m+

m 3

.

(10.12)

k

Solving equation (10.12) for the hanging mass, we have

( m+

m )= 3

k 4π

T2 .

(10.13)

Thus, equation (10.13) shows that the hanging mass is directly proportional to the square of the period of oscillation. We can use this fact and find the spring constant k experimentally by hanging various masses at the end of a vertical spring and measuring the period of oscillation of

these masses. A plot of the total hanging mass versus the square of the period of oscillation gives a straight line that goes through the origin, as in Fig. 10.3.

The slope of the straight-line equals

k 4π

. Therefore, k = 4 π

2

* Slope.

In this way, we can experimentally determine the spring constant for any spring. This method of deformation of the spring constant is known as the dynamic method.

Procedure: In this experiment we have to determine the spring constant k by two methods: a) by measuring the hanging weights and the corresponding elongation x of the spring and then determining k from the slope of the graph of the magnitude of the applied weight W versus the elongation x; b) by measuring the period of oscillation T of the vertical spring and corresponding hanging mass and then determining k from the slope of the graph of the total hanging mass versus the square of the period of oscillation T2.

Part A: The static method of the measurement of k

Part B: The dynamic method of the measurement of k

GRAPHS SPRING 1(Static Method) Elongation for spring

3 2.5 f(x) = 0.21 x − 0.03 2 1.5 1 0.5 0

2

4

6

8

10

Suspended weight Linear ()

Linear ()

12

14

SPRING 2(Static Method) Elongation for spring

3 2.5 f(x) = 0.13 x − 0.02

2 1.5 1 0.5 0

4

6

8

10

12

14

16

18

20

Suspended weight Linear ()

Linear ()

Linear ()

Total Mass

SPRING 1(Dynamic Method) 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 50

f(x) = 0 x + 0.02

100

150

200

Square of period Linear ()

Linear ()

250

300

SPRING 2(Dynamic Method) 0.8 0.7

Total Mass

0.6 0.5 0.4

f(x) = 0.01 x + 0.35

0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

Square of period Linear ()

Linear ()

CONCLUSION In this lab experiment, we studied Hooke’s law and determined the spring constant using the static and dynamic methods of measurements....