12 problems only
1. A human wave. During sporting events within large, densely packed stadiums, spectators will send a wave (or pulse) around the stadium (the figure). As the wave reaches a group of spectators, they stand with a cheer and then sit. At any instant, the width w of the wave is the distance from the leading edge (people are just about to stand) to the trailing edge (people have just sat down). Suppose a human wave travels a distance of 863 seats around a stadium in 55.7 s, with spectators requiring about 2.00 s to respond to the wave’s passage by standing and then sitting. What are (a) the wave speed v and (b) width w?
a. _____ seat/s
b. _____ seats
2. A sinusoidal transverse wave of wavelength 15.0 cm travels along a string in the positive direction of an x axis. The displacement y of the string particle at x = 0 is given in the figure as a function of time t. The scale of the vertical axis is set by ys = 2 cm. The wave equation is to be in the form of
y = ym sin(kx – ωt + φ).
(a) At t = 0, is a plot of y versus x in the shape of a positive sine function or a negative sine function? What are(b) ym, (c) k, (d) ω, (e) φ, (f) the sign in front of φ, and (g) the wave speed (speed of the wave along the string) and (h) What is the transverse velocity of the particle at x = 0 when t = 3.00 s?
3. The linear density of a string is 2.4 × 10-4 kg/m. A transverse wave on the string is described by the equation y = (0.046 m) sin[(1.8 m-1)x + (40 s-1)t] What are (a) the wave speed and (b) the tension in the string?
4. A standing wave pattern on a string is described by y(x, t) = 0.048 sin (7πx)(cos 56πt), where x and y are in meters and t is in seconds. For x ≥ 0, what is the location of the node with the (a) smallest, (b) second smallest, and (c) third smallest value of x? (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e) speed and (f) amplitude of the two traveling waves that interfere to produce this wave? For t ≥ 0, what are the (g) first, (h) second, and (i) third time that all points on the string have zero transverse velocity?
5. In the figure, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L = 1.1 m, linear density μ = 1.1 g/m, and the oscillator frequency f = 180 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if m = 2 kg (Give 0 if the mass cannot set up a standing wave)?
6. A 82.7 g wire is held under a tension of 150 N with one end at x = 0 and the other at x = 12.2 m. At time t = 0, pulse 1 is sent along the wire from the end at x = 12.2 m. At time t = 25.7 ms, pulse 2 is sent along the wire from the end at x = 0. At what position x do the pulses begin to meet?
7. Two identical traveling waves, moving in the same direction, are out of phase by π/4.0 rad. What is the amplitude of the resultant wave in terms of the common amplitude ym of the two combining waves? (Give the answer as the ratio of the total amplitude to the common amplitude.)
8. A solid sphere of uniform density has a mass of 9.4 × 104 kg and a radius of 1.5 m. What is the magnitude of the gravitational force due to the sphere on a particle of mass 3.0 kg located at a distance of (a) 5.6 m and (b) 1.2 m from the center of the sphere? (c) Write a general expression for the magnitude of the gravitational force on the particle at a distance r ≤ 1.5 m from the center of the sphere.
9. Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through its center. Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let FR be the magnitude of the gravitational force on the apple when it is located at the planet’s surface. How far from the surface (what multiple ofR) is there a point where the magnitude of the gravitational force on the apple is 0.3 FR if we move the apple (a)away from the planet and (b) into the tunnel?
10. A projectile is shot directly away from Earth’s surface. Neglect the rotation of the Earth. What multiple of Earth’s radius RE gives the radial distance (from the Earth’s center) the projectile reaches if (a) its initial speed is 0.747 of the escape speed from Earth and (b) its initial kinetic energy is 0.747 of the kinetic energy required to escape Earth? (Give your answers as unitless numbers.) (c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth?
11. In the figure, two spheres of mass m = 6.44 kg. and a third sphere of mass M form an equilateral triangle, and a fourth sphere of mass m4 is at the center of the triangle. The net gravitational force on that central sphere from the three other spheres is zero. (a) What is the value of mass M? (b) If we double the value of m4, what then is the magnitude of the net gravitational force on the central sphere?
12. A thin rod with mass M = 5.74 kg is bent in a semicircle of radius R = 0.626 m (the figure below). (a) What is its gravitational force (both magnitude and direction on a particle with mass m = 4.01×10-3 kg at P, the center of curvature? (b) What would be the force on the particle if the rod were a complete circle?
13. The three spheres in the figure, with masses mA = 75 g, mB = 10 g, and mC = 22 g, have their centers on a common line, with L = 23 cm and d = 5 cm. You move sphere B along the line until its center-to-center separation from C is d= 5 cm. How much work is done on sphere B (a) by you and (b) by the net gravitational force on B due to spheres Aand C?
14. Mountain pull. A large mountain can slightly affect the direction of “down” as determined by a plumb line. Assume that we can model a mountain as a sphere of radius R = 3.00 km and density (mass per unit volume) 2.6 × 103kg/m3. Assume also that we hang a 0.750m plumb line at a distance of 3R from the sphere’s center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere?