Virbations and Waves 1.1 Observe and find a pattern Try the following simple experiments and describe common patterns concerning the behavior of the block. (a) Fill in the table that follows. Experiment Record your observations. Tie a string to a small, heavy block and let the block hang freely. Now pull the block to the side and release it. Hang a heavy block from a spring, pull the block down, and release it. (b) Identify patterns common to both experiments. 1.2 Explain In Activity 1.1 you found that both the block on a string and the block on a spring had repeatable motion, either back and forth or up and down. The blocks moved about the place where they resided when not vibrating, that is, about the equilibrium position. Explain why each block returns to this equilibrium position, first moving in one direction and then a short time later in the opposite direction. To help your thinking, draw force diagrams for the block when on each side of the equilibrium position. V IV III II I x
1.3 Represent and reason The cart in the figure is attached to a special mass-less spring that can stretch and compress equally well. The cart and spring rest on a low-friction horizontal surface. The cart is pulled to position I and then released. It moves to position V, where it then reverses direction and returns again to position I. It repeats the motion. Represent with motion diagrams the cart s motion between the points indicated in the table that follows. Draw a motion diagram for motion between points I-III. Draw a motion diagram for motion between points III-V. Draw a motion diagram for motion between points V-III. Draw a motion diagram for motion between points III-I. 1.4 Represent and reason Represent with force diagrams the forces that other objects exert on the cart in Activity 1.2 at each point. Show only the horizontal force or forces that other objects exert on the cart. (The upward normal force of the surface on the cart s wheels and the downward gravitational force of the Earth on the cart cancel.) Draw the arrow lengths proportional to the magnitude(s) of the force(s) for each position. (a) Fill in the table that follows. diagram for point V. diagram for point IV. moving left: diagram for point III. moving left: diagram for point II. moving left: diagram for point I. moving right: moving right: moving right: (b) Do the diagrams depend on whether the cart was moving left or right? Explain. (c) Is the force description in the force diagrams consistent with the motion diagram description in Activity 1.3? For example, is the net horizontal force in the same direction as the acceleration? Give several specific examples. (d) At each position, compare the direction of the net force exerted by the spring on the cart and the cart s displacement from equilibrium when at that position.
1.5 Represent and reason (a) Construct five qualitative bar charts for the cart-spring system described in Activities 1.2 and 1.3 at the points described in the table that follows. point I. point II. point III. point IV. point V. (b) Do the charts depend on whether the cart is moving left when at a particular position or moving right? Explain. (c) How would the charts change if the surface had considerable friction? Explain. 1.6 Reason and explain Summarize the results of Activities 1.2-1.5 to describe and explain the motion of the cart. The description should include your observations, and the explanations should include reasoning based on force and energy analysis for the observed phenomena. Period (T) is a physical quantity that characterizes the time interval for one complete vibration.
1.7 Observe and find a pattern Hold one end of a slinky and have your partner hold the other. Stretch the slinky, so that it is about 3 to 4 m long. Do not lift it. Fill in the table that follows. (a) Keeping the slinky on the smooth surface and stretched along a straight line, give the end of the slinky in your hand a quick push along the axis of the slinky. Describe what you observe. (b) Sketch the Slinky at one instant of time during the propagation of the disturbance you created in part (a). (c) Indicate in words or draw how an individual Slinky ring in the middle of the Slinky moves with respect to the Slinky as the disturbance passes. (d) If you repeat the procedure described in part (a) but this time push more abruptly, does the disturbance move faster along the Slinky? How do you know? (e) If you push less abruptly than in part (a), does the speed of the disturbance change? How do you know? 1.8 Observe and find a pattern Keep the Slinky toy from 1.7 on the floor and fastened securely at one end. Again grasp the free end of the Slinky and stretch it so that the Slinky is about 3 to 4 m long. Fill in the table that follows. (a) Give the end of the Slinky in your hand an abrupt sideways shake, perpendicular to the Slinky (see the top-view illustration), all the while keeping the Slinky on the smooth surface.. Describe what you observe.
(b) Sketch the Slinky at one instant of time during the propagation of the disturbance you created in part (a). (c) Indicate in words or draw how an individual Slinky ring in the middle of the Slinky moves with respect to the Slinky as the disturbance passes. (d) If you repeat the procedure described in part a, but this time make a larger abrupt sideways shake, does the disturbance move faster along the Slinky? (e) If you make a smaller abrupt sideways shake than in part a, does the speed of the disturbance change? Pulses: A longitudinal pulse in this context is a pulse in which individual coils of a Slinky move in the direction of the propagation of the disturbance. A transverse pulse is a pulse in which the individual coils move in the direction perpendicular to the propagation of the pulse on the slinky. 1.9 Reason and explain Small balls of mass m are connected with small springs of spring constant k (a short section is shown in the figure). m k m k m k m k m The balls and springs rest on a smooth, frictionless surface. Imagine that you vibrate one end of this chain of balls and springs back and forth, parallel to the axis of the chain, causing a wave disturbance that moves along the spring-mass chain at some speed v. (a) Do you think the speed of the waves along the chain depends on the spring constant k? If so, do you think the speed is greater or less for greater spring constants? Explain. (b) Do you think the speed of the waves along the chain depends on the mass m of the balls? If so, do you think the speed is greater or less for greater mass? Explain. (c) By analogy, identify two properties of stretched strings (for example, a violin, guitar, or piano string) that might affect the speed of waves on the strings.
1.1 Design an experiment Design an experiment to determine if a transverse pulse or a longitudinal pulse moves with greater speed along a Slinky. Sketch the experimental set-up. Explain in words how you will measure the speed for each kind of pulse. List quantities that you will measure and quantities that you will calculate. List your assumptions and experimental uncertainties. To be measured: To be calculated: Assumptions: Perform the experiment; record the results and describe your conclusion. Uncertainties: 1.11 Reason The speed of a wave depends on properties of the medium through which the wave travels. The speed v can be determined in a different way if you know the wavelength λ of the wave and the period T of the vibration, then v = λ/t. (a) Explain why this equation makes sense. (b) Rewrite the expression in (a) using the wave frequency f = 1/T to show that speed, frequency, and wavelength are related as follows: v = f λ. 1.12 Reason The frequency f of a wave equals 1/T (one vibration in a time interval T). (a) Explain why this makes sense. (b) Suppose that there are 1 vibrations in 5 s. What is the frequency of such a wave and what is its period? (c) If the wave travels at speed 4. m/s, determine the wavelength of the wave. (d) Show that λ = v/f. **Textbook: Sections 14.1 and 14.2. Make sure you are comfortable with the vocabulary.