Supplementary Problems for Topics III Solutions 1. An empty steel drum of mass M, height h and cross-section area A is floating partially submerged as shown in water. The bottom of the drum is at depth d below the surface. Express all answers in terms of the given quantities, g, and the density ρ of water. a. What is d? b. Now an external force of magnitude F is exerted on the top of the drum, pushing it further down into the water by an extra distance x0. Express F in terms of x0 and the other quantities. c. Prove that if the external force is suddenly removed the drum will execute simple harmonic motion in vertical oscillations. [How does the net force depend on the vertical displacement x of the drum from its equilibrium position?] d. Find the angular frequency of those oscillations. a. The buoyant force must equal the weight of the drum, so Mg = Ad!g, or d = M /!A. b. The applied force must be balanced by the extra buoyant force, so F =!Ax0g. c. If the drum is distance x deeper than its equilibrium depth, the extra buoyant force is !Axg, upward (opposite to the displacement). Thus we have a restoring force, opposite to the displacement, proportional to the amount of the displacement. That is what give rise to SHM. d. The standard form for the restoring force is F = !M"2x. Since we have F = !"Agx, we see that !2="Ag/ M. d2. An aspirator is a device that mixes small amounts of a fluid (such as liquid fertilizer) with water, so that it can be sprayed over a large area. Shown is such a device. Water from a hose (on the left) flows through a constriction, where a vertical tube introduces the fluid (at negligible speed) into the rapid water flow. The height through which the fluid must rise is h as shown. If the fluid-water mixture exits into the air at speed v0, what must be the minimum ratio A/a of the cross-section areas of the two regions? Assume the fluid and water have the same density ρ. [Find the water pressure in the constriction.] a. Use Bernoulli, comparing the point in the constriction with the point where the water enters the air: Pa+12!va2+!gy = P0+12!v02+!gy. (The points are at the same height, which we call y.) This gives va2= v02+ 2(P0! Pa)/". b. Now compare the top of the fluid in the fluid tank with the point in the constriction where the fluid enters the water. The fluid moves very slowly so its speed can be neglected. P0= Pa+!gh, or P0! Pa="gh. Thus we have va2= v02+ 2gh. c. Finally, use continuity for the flow of the water: ava= Av0. Then we have (A/ a)2v02= v02+ 2gh, or (A/ a)2= 1 + 2gh/v02. a h A v 03. An ideal massless spring of stiffness k has its lower end attached to a table, and its upper end attached to a plate of mass M. On top of the plate rests a block of mass m. a. Through what vertical distance does the weight of the plate and block compress the spring when the system is at rest? b. Now an external force pushes the block and plate downward an extra small distance A, compressing the spring further by that amount. The external force is then suddenly removed, allowing the system to oscillate vertically. What is the angular frequency ω of the oscillation? c. What is the maximum value of A for which the block will not separate from the plate during the oscillations? a. The spring supports the combined weight of the table and the block, so kd = (M + m)g, and d = (M + m)g/ k. b. As long as the block remains in contact with the table, the oscillating mass is M + m and !2= k /(M + m). c. The block cannot accelerate downward at a rate greater than g, while the table’s maximum downward acceleration is !2A, so we require !2A " g, or Amax= g/!2. M m k4. A block of mass m is attached as shown to a spring. This system is oscillating with amplitude A0 and maximum speed v0 on a frictionless floor. At the instant when the attached block is at its furthest position to the right (momentarily at rest) it is struck from the right by a second block of mass m, moving with speed v0. After the collision, the second block sticks to the first, and the combined system now oscillates with amplitude A and maximum speed v. a. What is the ratio of the final and initial angular frequencies of the oscillation? b. What is the ratio of the final and initial amplitudes A/A0? c. What is the ratio of the final and initial maximum speeds v/v0? [All answers are numerical. Leave numbers such as  2 in that form.] a. The new oscillating mass is 2m, so !2= k /2m, and !/!0= 1/2. b. The total energy initially is both 12mv02 and 12kA02, so v02= (k /m)A02. Now we need to find the final total energy. Conservation of momentum in the collision gives mv0= (2m)v1, so v1= v0/2. Here v1 is not the maximum speed in the final situation, but the speed just after the collision when the spring is still stretched by the amount A0. (Maximum speed occurs when the spring is unstretched.) The total energy is E =12(2m)v12+12kA02=14mv02+12kA02. In tems of the final amplitude, E =12kA2. Expressing v0 in terms of A0, we find 12kA2=34kA02, so A / A0= 3/2. c. The final energy is also E =12(2m)v2. Expressing A0 in terms of v0 we find mv2=14mv02+12mv02=34mv02, and v /v0= 3/4.5. The loudness of the sound from a rock band is 138 db at distance 10 m from the stage. How far will a listener who values his hearing move away from the stage so that the loudness is a tolerable 120 db? We have 138 = 10log(I1/I0) and 120 = 10log(I2/I0), so 18 = 10log(I1/I2). This gives I1/I2= 101.8! 63. Since I(x) ! x!2 we have x22/100 = 63, or x2= 6300 ! 79 m. 6. A child blowing a whistle of frequency f0 is running away from a vertical wall and toward a stationary listener. The child’s speed is 3.4 m/s, which is 1% of the speed of sound. The listener hears 4 beats/s produced by the sound directly from the whistle and that reflected from the wall. Find f0. For the frequency heard directly from the whistle: f1= f011 ! 0.01. For the frequency reflected from the wall: f2= f011 + 0.01 The beat frequency is fB= f1! f2, so we have 4 = f011 ! 0.01!11 + 0.01"#$%&'( 0.02 f0, and f0! 200 Hz. 7. Three successive harmonics in the sound emitted by an organ pipe have frequencies 1000, 1400 and 1800 Hz. a. What is the fundamental frequency? b. Is the pipe open at both ends or closed at one end? Explain. c. As a crude approximation, take the speed of sound to be 300 m/s. In that