Chapter 4: Kinematics in Two Dimensions

You are asked to consult for the city’s research hospital, where a group of doctors is investigating the bombardment of cancer tumors with high-energy ions. As figure shows, ions are fired directly toward the center of the tumor at speeds of $5.0\times {10}^6\mathrm{m}/\mathrm{s}$. To cover the entire tumor area, the ions are deflected sideways by passing them between two charged metal plates that accelerate the ions perpendicular to the direction of their initial motion. The acceleration region is $5\mathrm{cm}$long, and the ends of the acceleration plates are $1.5\mathrm{m}$from the target. What sideways acceleration is required to deflect an ion $2\mathrm{cm}$to one side?

Ships A and B leave port together. For the next two hours, ship A travels at $20\mathrm{mph}$in a direction $30°$west of north while ship B travels $20°$east of north at $25\mathrm{mph}$.

a. What is the distance between the two ships two hours after they depart?

b. What is the speed of ship A as seen by ship B?

While driving north at $25\mathrm{m}/\mathrm{s}$ during a rainstorm you notice that the rain makes an angle of $38°$ with the vertical. While driving back home moments later at the same speed but in the opposite direction, you see that the rain is falling straight down. From these observations, determine the speed and angle of the raindrops relative to the ground.

You’ve been assigned the task of using a shaft encoder a device that measures the angle of a shaft or axle and provides a signal to a computer to analyze the rotation of an engine crankshaft under certain conditions. The table lists the crankshaft’s angles over a $0.6\mathrm{s}$interval.

Is the crankshaft rotating with uniform circular motion? If so, what is its angular velocity in rpm? If not, is the angular acceleration positive or negative?

A circular track has several concentric rings where people can run at their leisure. Phil runs on the outermost track with radius r_P while Annie runs on an inner track with radius r_A = 0.80r_P. The runners start side by side, along a radial line, and run at the same speed in a counterclockwise direction. How many revolutions has Annie made when Annie’s and Phil’s velocity vectors point in opposite directions for the first time?

A typical laboratory centrifuge rotates at $4000\mathrm{rpm}$. Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations.

a. What is the acceleration at the end of a test tube that is $10\mathrm{cm}$from the axis of rotation?

b. For comparison, what is the magnitude of the acceleration a test tube would experience if dropped from a height of $1.0\mathrm{m}$and stopped in a $1.0\mathrm{ms}$long encounter with a hard floor?

Astronauts use a centrifuge to simulate the acceleration of a rocket launch. The centrifuge takes $30\mathrm{s}$to speed up from rest to its top speed of $1$rotation every $1.3\mathrm{s}$. The astronaut is strapped into a seat $6.0\mathrm{m}$from the axis.

a. What is the astronaut’s tangential acceleration during the first $30\mathrm{s}$?

b. How many g’s of acceleration does the astronaut experience when the device is rotating at top speed? Each $9.8\mathrm{m}/{\mathrm{s}}^2$of acceleration is$1\mathrm{g}$.

A computer hard disk $8.0\mathrm{cm}$in diameter is initially at rest. A small dot is painted on the edge of the disk. The disk accelerates at $600\mathrm{rad}/{\mathrm{s}}^2$for $\frac{1}{2}\mathrm{s}$, then coasts at a steady angular velocity for another $\frac{1}{2}\mathrm{s}$.

a. What is the speed of the dot at $\mathrm{t}=1.0\mathrm{s}$?

b. Through how many revolutions has the disk turned?

A high-speed drill rotating ccw at $2400\mathrm{rpm}$comes to a halt in $2.5\mathrm{s}$.

a. What is the magnitude of the drill’s angular acceleration?

b. How many revolutions does it make as it stops?

Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The radius of the earth is 6.37 * 10⁶ m, and the altitude of a geosynchronous orbit is 3.58 * 10⁷ m (≈22,000 miles). What are
(a) the speed and
(b) the magnitude of the acceleration of a satellite in a geosynchronous orbit?