*12 problems only*

*Ass 25*

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 (7*πx*)(cos 56*πt*), 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.)

Ass 26

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 of*R*) 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 *m*4 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 *m*4, 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 *A*and *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 3*R* 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?

15. In the figure, a particle of mass *m1* = 0.39 kg is a distance *d* = 41 cm from one end of a uniform rod with length *L*= 4.5 m and mass *M* = 1.1 kg. What is the magnitude of the gravitational force

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