91Ó°ÊÓ

A$\mathbf{}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{kg}$rock has a horizontal velocity of magnitude $\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}$when it is at point P in fig $\mathbf{E}\mathbf{10}\mathbf{.}\mathbf{35}$. (a) At this instant, what are the magnitude and direction of its angular momentum relative to point $\mathbf{O}$? (b) If the only force acting on the rock is its weight, what is the rate of change (magnitude and direction) of its angular momentum at this instant?

Consider the ring-shaped body of Fig. E13.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Useto find the magnitude and direction of the force on the particle (see Section 7.4). (d) Show that your answer to part (c) reduces to the expected result when x is much larger than a. (e) What are the values of U andwhen? Explain why these results make sense.

Vector$\stackrel{\mathbf{→}}{\mathbf{A}}$ S is 2.80 cm long and is$\mathbf{60}\mathbf{°}$ above the x-axis in the first quadrant. Vector$\stackrel{\mathbf{→}}{\mathbf{B}}$ is 1.90 cm long and is$\mathbf{60}\mathbf{°}$ below the x-axis in the fourth quadrant (Fig. E1.35). Use components to find the magnitude and direction of (a) role="math" localid="1663949431128" $\stackrel{\mathbf{→}}{\mathbf{A}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{B}}$; (b) $\stackrel{\mathbf{→}}{\mathbf{A}}-\stackrel{\mathbf{→}}{\mathbf{B}}$; (c) $\stackrel{\mathbf{→}}{\mathbf{B}}-\stackrel{\mathbf{→}}{\mathbf{A}}$. In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative

Basketball player Darrell Griffith is on record as attaining a standing vertical jump of 1.2 m(4 ft). (This means that he moved upward by 1.2m after his feet left the floor.) Griffith weighed 890 N(200lb). (a) What was his speed as he left the floor? (b) If the time of the part of the jump before his feet left the floor was 0.300 s, what was his average acceleration (magnitude and direction) while he pushed against the floor? (c) Draw his free-body diagram. In terms of the forces on the diagram, what was the net force on him? Use Newton’s laws and the results of part (b) to calculate the average force he applied to the ground.

In a head-on auto collision, passengers who are not wearing seat belts may be thrown through the windshield. Use Newton’s laws of motion to explain why this happens.

(a) In which direction should the motorboat in Exercise 3.35 head to reach a point on the opposite bank directly east from your starting point? (The boat’s speed relative to the water remains 4.2 m/s) (b) What is the velocity of the boat relative to the earth? (c) How much time is required to cross the river.

A mass is oscillating with amplitude A at the end of a spring. How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?

You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of $\mathbf{395}.\mathbf{0}\text{ }\mathbf{N}$in the rope. If Sneezy hangs from a similar rope while delivering presents at the earth’s equator, what will the tension in it be? (Recall that the earth is rotating about an axis through its north and south poles.) Consult Appendix F and start with a free-body diagram of Sneezy at the equator.

Water runs into a fountain, filling all the pipes, at a steady rate of $\mathbf{0}\mathbf{.}\mathbf{750}{\mathbf{m}}^{\mathbf{3}}\mathbf{/}\mathbf{s}$. (a) How fast will it shoot out of a hole 4.50 cm in diameter? (b) At what speed will it shoot out if the diameter of the hole is three times as large?

36) A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at t = 0, the wheel turns through 8.20 revolutions in 12.0 s. At t = 12.0 s the kinetic energy of the wheel is 36.0 J. For an axis through its center, what is the moment of inertia of the wheel?