Objectives 1. To determine the spring constant k by stretching the spring and applying Hooke’s Law; 2. To verify that the period of vibration of a body on a spring is independent of the amplitude, and is given by: (1) T = 2 π √ m k 3. To measure the period T of a sample pendulum as a function of the length L of the pendulum, and to verify that it is given by: (2) T = 2 π √ L g Introduction and Background I. Spring When a body of mass m is suspended on a coil spring with constant k , and if the spring is stretched or compressed from its equilibrium position through a displacement s , the spring exerts on the body a force that is proportional to the displacement and given by Hooke’s law (when within elastic limit): (3) ⃗ F =− k ⃗ s,of which magnitudeisF = ks We call F a restoring force. The negative sign indicates that the force direction is opposite to the direction of the displacement. If the spring is hung vertically, when the body is stationary, according to Newton’s First Law we know that (4) F = mg Combine equations (3) and (4), we have: (5) mg = ks In Part I of the experiment, we will use equation (5) to determine the spring constant k . When the body is stationary, we call the position of the system (spring + body) the equilibrium position of the system. If the body of mass m is now displaced from the equilibrium position of the system (spring + body) a distance x , there will be a net force acting on the body. According to
Newton’s Second Law, this net force will cause the body vibrating back and forth around the system equilibrium position. If the vibration amplitude of the body is relatively small, its motion is Simple Harmonic and its period of oscillation T is given by equation (1). Notice that the oscillation period T is independent of the vibration amplitude. II. Simple Pendulum Simple pendulum is also an example of simple harmonic motion. For a small displacement angle, the period T is related to length L through equation (2). Fir the plot of L vs. T 2 with a regression line, gravitational acceleration g could be deduced from the slope. Experimental Procedures Part I: Determine the spring constant k by stretching the spring and applying Hooke’s Law 1. With the lab equipment provided in the counter tray, clamp a one- or two-meter stick in a vertical position. Hang the spring at the groove of the rod and carefully record the position y 0 of the bottom of the spring. Then hang the 50 g weight hanger at the bottom of the spring. 2. Add 100g weight to the weight hanger and record the position y 1 of the bottom of the spring.