EXPERIMENT 1: HOOKE’S LAW AND OSCILLATIONS 1.1. OBJECTIVES The objective of this experiment is to determine the spring constant of a springs using (i) the static method (Hooke’s Law); and (ii) the dynamics method (oscillations) 1.2. APPARATUS Spiral spring, pointer, mass hanger, rigid stand and clamp, metre rule, slotted masses, triple balance and stop watch. 1.3. THEORY When an external force 𝐹𝑒𝑥𝑡 is applied to a spring (or indeed any elastic material), a force is developed within the spring that attempts to restore the original configuration of the spring in accordance with Newton’s third law. This force is called the restoring force. Robert Hooke showed that if the spring is stretched out by an amount 𝑥, then the restoring force is linearly proportional to 𝑥, provided that the elastic limit of the spring is not exceeded. This remarkable behavior of the spring is known as Hooke’s law and is expressed by the equation 𝐹𝑟 = −𝑘𝑥 Eqn (1) where the proportionality constant 𝑘 is called the spring (stiffness) constant. As the name suggests, the spring constant is a property of a spring that represents the ‘stiffness’ of that particular spring. The negative sign in Equation (I) signifies the fact that the restoring force is in a direction that is opposite to the applied force. In the first part of this experiment, we are going to verify Hooke’s law by demonstrating that the magnitude of the restoring force (given by Equation 1) equals that of the applied external force. In our case, the external force will be provided by the force of gravity due to a mass 𝑚 attached to the end of the spring. The direction of this force is (of course) downwards, and it must counter the restoring force of the spring which acts upwards. At the equilibrium point (where the two forces balance), we have 𝐹𝑔 − 𝐹𝑟 = 0 Eqn (2) by Newton’s third law. This can be written as 𝑚𝑔 − 𝑘𝑥 = 0 Or 𝑚𝑔 = 𝑘𝑥 Eqn (3) Equation (3) tells us that a plot of the applied force (mg) against the extension 𝑥 would give us a straight line whose intercept is zero and slope the spring constant 𝑘. This method of determining the spring constant is called the static method because the measurements are performed in a static situation, when the mass is in equilibrium (not moving). Consider a situation where the mass 𝑚 is in equilibrium, i.e., the upwards restoring force is equal to the downwards weight. If now the spring is stretched beyond the equilibrium point by pulling it down slightly and then releasing it, the mass will accelerate upward because the restoring force due to the spring is larger than the force of gravity pulling down. After release it will pass through the equilibrium point and continue to move upward. Once above the equilibrium position, gravity will start to exceed the force pulling upward due to the spring and acceleration will be directed downward. The result of this is that the mass will oscillate around the equilibrium position. The oscillations will proceed with a characteristic period, T, which is the time it takes for the spring to complete one oscillation, or the time necessary for the mass to return to the point where the cycle starts repeating. As we shall see in class, the period of oscillation depends on the spring constant and the total attached mass via, Eqn (4) Note here that 𝑚 is the total mass of the hanger, applied load and the spring. Squaring Equation (4) both sides gives Eqn (5) Thus a graph of 𝑇2 against 𝑚 gives 4𝜋2⁄𝑘 as slope, from which can be obtained. This method of determining the spring constant is called the dynamic method, since measurements are done when the spring is in ‘dynamic equilibrium’. Note here that 𝑚 is the total mass of the load including the mass hanger and spring. 1.4. PROCEDURE Measure and record the total mass of the mass hanger and spring using a triple balance. Then set up the apparatus as shown in Figure 1 PART 1 : 1. Note the reading of the pointer when no mass is attached to the spring. 2. Add slotted masses to the mass hanger, recording the position of the pointer at each stage. You may decide what increments of masses are suitable to give you enough data points (not less than five). Make sure not to load the spring beyond its elasticity limit. Consult your lab assistants whenever unsure. 3. The extension 𝑥 at each stage is obtained by subtracting the zero-load reading (obtained in Step 1) from the pointer reading when a certain load has been mounted. 4. After reaching the maximum desirable load, start removing masses, taking the pointer readings at each stage as described in Step 3. 5. Record your readings in a table such as the one shown below: Zero-load reading: ……………… (cm) S/N Load m (g) 1 2 3 4 5 6 Pointer Readings Increasing Decreasing Load (cm) Load (cm) Extension of Spring Increasing Decreasing Load (cm) Load (cm) Mean Extension 𝑥 (cm) 1.5. DATA ANALYSIS : Plot a graph of extension 𝑥 versus load 𝐹𝑔 = 𝑚𝑔, choosing the correct origin and scales. Determine the gradient (slope) of the graph and use this to calculate the spring constant 𝑘. Give the value of the calculated 𝑘 in S.I. units. NOTE: Since every measurement you make is not exact, but has some experimental uncertainty, do not expect your data points to lie exactly on a smooth curve or a perfectly straight line. Question : Explain why it is necessary to take the readings while both increasing and decreasing the load. PART 2 : 1. Remove the pointer from the mass hanger. 2. Add a load 𝑚 to the hanger and set it in vertical vibration by giving it a small additional downward displacement. Obtain the time taken for 10 vibrations three times. 3. Repeat the measurements with different loads. You are free to choose the loads that you feel are suitable. However, the loads should be in such a way that the coils do not touch when the spring is compressed during the oscillations. 4. Record your readings in a table such as the one shown below. Mass of hanger + spring: M =………….g S/N Load m (g) Time for 10 Total Load oscillation (m + M) (g) t1 (s) t2 (s) complete Mean time Period of Oscillation, for 10 oscillations T (s) T2 (s2) t3 (s) (s) 1 2 3 4 5 6 Plot a graph of T2 on the y-axis against (M + m) on the x-axis and determine the slope. Does the graph confirm your theoretical expectations? Compare the values of 𝑘 found in parts (i) and (ii), and comment.