Determine whether the following trajectory lies on a circle. If so, find the radius of the circle and show that the position vector and velocity vector are everywhere orthogonal. r(t) = (2 sint+ V5 cos t 5 sint-2 cos t) for 0sts 2n Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The trajectory lies on a circle with radius O B. The trajectory does not lie on a circle. What is the velocity vector? v(t) = (Type an exact answer, using radicals as needed.) Find the dot product r•v to show that the position and velocity vector are orthogonal. r•v =

Gear Trains Figure 8 shows a single-stage gear train. Gear trains are used in many products, such as clocks and automotive transmissions, to reduce or increase the angular velocity of a component. The size of each gear is measured by the number of teeth rather than the radius. Suppose the first gear has n1 and the second gear has n2 teeth. Because the spacing of the teeth is the same for both gears, the ratio of their radii will be equivalent to the corresponding ratio of the number of teeth. When two gears are meshed together, they share the same linear velocity. If 1 and 2 are the angular velocities of the first and second gears, respectively, then v2=v1r22=r112=r1r212=n1n21 The first gear in a single-stage gear train has 42 teeth and an angular velocity of 2 revolutions per second. The second gear has 7 teeth. Find the angular velocity of the second gear.

Gear Trains Figure 8 shows a single-stage gear train. Gear trains are used in many products, such as clocks and automotive transmissions, to reduce or increase the angular velocity of a component. The size of each gear is measured by the number of teeth rather than the radius. Suppose the first gear has n1 and the second gear has n2 teeth. Because the spacing of the teeth is the same for both gears, the ratio of their radii will be equivalent to the corresponding ratio of the number of teeth. When two gears are meshed together, they share the same linear velocity. If 1 and 2 are the angular velocities of the first and second gears, respectively, then v2=v1r22=r112=r1r212=n1n21 The second gear in a single-stage gear train has 6 teeth and an angular velocity of 90 revolutions per minute. The first gear has 54 teeth. Find the angular velocity of the first gear.

Express the lengths a and b shown in the figure in terms of .

Find the angle between u=2,1 and v=1,3.