Chapter 7

The Cartesian coordinates of a point on a circle are \((1.5 \mathrm{~m}, 2.0 \mathrm{~m}) .\) What are the polar coordinates \((r, \theta)\) of this point?

The polar coordinates of a point are \(\left(5.3 \mathrm{~m}, 32^{\circ}\right)\). What are the point's Cartesian coordinates?

Convert the following angles from degrees to radians, to two significant figures: (a) \(15^{\circ},(\mathrm{b}) 45^{\circ},(\mathrm{c}) 90^{\circ},\) and (d) \(120^{\circ}\)

Convert the following angles from radians to degrees: (a) \(\pi / 6 \mathrm{rad}\) (b) \(5 \pi / 12 \mathrm{rad}\) (c) \(3 \pi / 4 \mathrm{rad},\) and (d) \(\pi\) rad.

Express the following angles in degrees, radians, - Express the following angles in degrees, radians, and/or revolutions (rev) as appropriate: (a) \(105^{\circ}\) (b) \(1.8 \mathrm{rad},\) and \((\mathrm{c}) 5 / 7\) rev.

You measure the length of a distant car to be subtended by an angular distance of \(1.5^{\circ} .\) If the car is actually \(5.0 \mathrm{~m}\) long, approximately how far away is the car?

The hour, minute, and second hands on a clock are \(0.25 \mathrm{~m}, 0.30 \mathrm{~m},\) and \(0.35 \mathrm{~m}\) long, respectively. What are the distances traveled by the tips of the hands in a 30-min interval?

A car with a 65-cm-diameter wheel travels \(3.0 \mathrm{~km}\). How many revolutions does the wheel make in this distance?

Two gear wheels with radii of \(25 \mathrm{~cm}\) and \(60 \mathrm{~cm}\) have interlocking teeth. How many radians does the smaller wheel turn when the larger wheel turns 4.0 rev?

You ordered a 12-in. pizza for a party of five. For the pizza to be distributed evenly, how should it be cut in triangular pieces?

A bicycle wheel has a small pebble embedded in its tread. The rider sets the bike upside down, and accidentally bumps the wheel, causing the pebble to move through an arc length of \(25.0 \mathrm{~cm}\) before coming to rest. In that time, the wheel spins \(35^{\circ} .\) (a) The radius of the wheel is therefore (1) more than \(25.0 \mathrm{~cm},(2)\) less than \(25.0 \mathrm{~cm},\) (3) equal to \(25.0 \mathrm{~cm}\) (b) Determine the radius of the wheel.

At the end of her routine, an ice skater spins through 7.50 revolutions with her arms always fully outstretched at right angles to her body. If her arms are \(60.0 \mathrm{~cm}\) long, through what arc length distance do the tips of her fingers move during her finish?

(a) Could a circular pie be cut such that all of the wedge-shaped pieces have an arc length along the outer crust equal to the pie's radius? (b) If not, how many such pieces could you cut, and what would be the angular dimension of the final piece?

Electrical wire with a diameter of \(0.50 \mathrm{~cm}\) is wound on a spool with a radius of \(30 \mathrm{~cm}\) and a height of \(24 \mathrm{~cm}\). (a) Through how many radians must the spool be turned to wrap one even layer of wire? (b) What is the length of this wound wire?

A yo-yo with an axle diameter of \(1.00 \mathrm{~cm}\) has a \(90.0-\mathrm{cm}\) length of string wrapped around it many times in such a way that the string completely covers the surface of its axle, but there are no double layers of string. The outermost portion of the yo-yo is \(5.00 \mathrm{~cm}\) from the center of the axle. (a) If the yo-yo is dropped with the string fully wound, through what angle does it rotate by the time it reaches the bottom of its fall? (b) How much arc length has a piece of the yo-yo on its outer edge traveled by the time it bottoms out?

A computer DVD-ROM has a variable angular speed from 200 rpm to 450 rpm. Express this range of angular speed in radians per second.

A race car makes two and a half laps around a circular track in \(3.0 \mathrm{~min}\). What is the car's average angular speed?

What are the angular speeds of the (a) second hand, (b) minute hand, and (c) hour hand of a clock? Are the speeds constant?

What is the period of revolution for (a) a 9500-rpm centrifuge and (b) a 9500-rpm computer hard disk drive?

Determine which has the greater angular speed: particle \(A\), which travels \(160^{\circ}\) in \(2.00 \mathrm{~s}\), or particle \(\mathrm{B}\), which travels \(4 \pi\) rad in \(8.00 \mathrm{~s}\).

The tangential speed of a particle on a rotating wheel is \(3.0 \mathrm{~m} / \mathrm{s}\). If the particle is \(0.20 \mathrm{~m}\) from the axis of rotation, how long will the particle take to make one revolution?

A merry-go-round makes 24 revolutions in a 3.0 -min ride. (a) What is its average angular speed in rad/s? (b) What are the tangential speeds of two people \(4.0 \mathrm{~m}\) and \(5.0 \mathrm{~m}\) from the center, or axis of rotation?

The Earth rotates on its axis once a day and revolves around the Sun once a year. (a) Which is greater, the rotating angular speed or the revolving angular speed? Why? (b) Calculate both angular speeds in \(\mathrm{rad} / \mathrm{s}\).

A little boy jumps onto a small merry-go-round (radius of \(2.00 \mathrm{~m}\) ) in a park and rotates for 2.30 s through an arc length distance of \(2.55 \mathrm{~m}\) before coming to rest. If he landed (and stayed) at a distance of \(1.75 \mathrm{~m}\) from the central axis of rotation of the merry-go-round, what was his average angular speed and average tangential speed?

The driver of a car sets the cruise control and ties the steering wheel so that the car travels at a uniform speed of \(15 \mathrm{~m} / \mathrm{s}\) in a circle with a diameter of \(120 \mathrm{~m}\). (a) Through what angular distance does the car move in \(4.00 \mathrm{~min} ?\) (b) What arc length does it travel in this time?

In a noninjury, noncontact skid on icy pavement on an empty road, a car spins 1.75 revolutions while it skids to a halt. It was initially moving at \(15.0 \mathrm{~m} / \mathrm{s}\), and because of the ice it was able to decelerate at a rate of only \(1.50 \mathrm{~m} / \mathrm{s}^{2}\). Viewed from above, the car spun clockwise. Determine its average angular velocity as it spun and slid to a halt.

An Indy car with a speed of \(120 \mathrm{~km} / \mathrm{h}\) goes around a level, circular track with a radius of \(1.00 \mathrm{~km} .\) What is the centripetal acceleration of the car?

A wheel of radius \(1.5 \mathrm{~m}\) rotates at a uniform speed. If a point on the rim of the wheel has a centripetal acceleration of \(1.2 \mathrm{~m} / \mathrm{s}^{2},\) what is the point's tangential speed?

A rotating cylinder about \(16 \mathrm{~km}\) long and \(7.0 \mathrm{~km}\) in diameter is designed to be used as a space colony. With what angular speed must it rotate so that the residents on it will experience the same acceleration due to gravity as on Earth?

An airplane pilot is going to demonstrate flying in a tight vertical circle. To ensure that she doesn't black out at the bottom of the circle, the acceleration must not exceed \(4.0 g .\) If the speed of the plane is \(50 \mathrm{~m} / \mathrm{s}\) at the bottom of the circle, what is the minimum radius of the circle so that the \(4.0 \mathrm{~g}\) limit is not exceeded?

Imagine that you swing about your head a ball attached to the end of a string. The ball moves at a constant speed in a horizontal circle. (a) Can the string be exactly horizontal? Why or why not? (b) If the mass of the ball is \(0.250 \mathrm{~kg}\), the radius of the circle is \(1.50 \mathrm{~m}\), and it takes 1.20 s for the ball to make one revolution, what is the ball's tangential speed? (c) What centripetal force are you imparting to the ball via the string?

A car with a constant speed of \(83.0 \mathrm{~km} / \mathrm{h}\) enters a circular flat curve with a radius of curvature of \(0.400 \mathrm{~km} .\) If the friction between the road and the car's tires can supply a centripetal acceleration of \(1.25 \mathrm{~m} / \mathrm{s}^{2},\) does the car negotiate the curve safely? Justify your answer.

In performing a "figure \(8^{\prime \prime}\) maneuver, a figure skater wants to make the top part of the 8 approximately a circle of radius \(2.20 \mathrm{~m} .\) He needs to glide through this part of the figure at approximately a constant speed, taking \(4.50 \mathrm{~s} .\) His skates digging into the ice are capable of providing a maximum centripetal acceleration of \(3.25 \mathrm{~m} / \mathrm{s}^{2}\). Will he be able to do this as planned? If not, what adjustment can he make if he wants this part of the figure to remain the same size (assume the ice conditions and skates don't change)?

A light string of length of \(56.0 \mathrm{~cm}\) connects two small square blocks, each with a mass of \(1.50 \mathrm{~kg} .\) The system is placed on a slippery (frictionless) sheet of horizontal ice and spun so that the two blocks rotate uniformly about their common center of mass, which itself does not move. They are supposed to rotate with a period of \(0.750 \mathrm{~s}\). If the string can exert a force of only \(100 \mathrm{~N}\) before it breaks, determine whether this string will work.

A jet pilot puts an aircraft with a constant speed into a vertical circular loop. (a) Which is greater, the normal force exerted on the seat by the pilot at the bottom of the loop or that at the top of the loop? Why? (b) If the speed of the aircraft is \(700 \mathrm{~km} / \mathrm{h}\) and the radius of the circle is \(2.0 \mathrm{~km}\), calculate the normal forces exerted on the seat by the pilot at the bottom and top of the loop. Express your answer in terms of the pilot's weight.

For a scene in a movie, a stunt driver drives a \(1.50 \times 10^{3} \mathrm{~kg} \mathrm{SUV}\) with a length of \(4.25 \mathrm{~m}\) around a circular curve with a radius of curvature of \(0.333 \mathrm{~km}\) (vFig. 7.34\()\). The vehicle is to be driven off the edge of a gully \(10.0 \mathrm{~m}\) wide, and land on the other side \(2.96 \mathrm{~m}\) below the initial side. What is the minimum centripetal acceleration the SUV must have in going around the circular curve to clear the gully and land on the other side?

Consider a simple pendulum of length \(L\) that has a small mass (the bob) of mass \(m\) attached to the end of its string. If the pendulum starts out horizontally and is released from rest, show that (a) the speed at the bottom of the swing is \(v_{\max }=\sqrt{2 g L}\) and \((b)\) the tension in the string at that point is three times the weight of the bob, or \(T_{\max }=3 m g .\) [Hint: Use conservation of energy to determine the speed at the bottom and centripetal force ideas and a free-body diagram to determine the tension at the bottom.

A CD originally at rest reaches an angular speed of \(40 \mathrm{rad} / \mathrm{s}\) in \(5.0 \mathrm{~s}\). (a) What is the magnitude of its angular acceleration? (b) How many revolutions does the CD make in the \(5.0 \mathrm{~s} ?\)

A merry-go-round accelerating uniformly from rest achieves its operating speed of 2.5 rpm in 5 revolutions. What is the magnitude of its angular acceleration?

A flywheel rotates with an angular speed of 25 rev \(/\) s. As it is brought to rest with a constant acceleration, it turns 50 rev. (a) What is the magnitude of the angular acceleration? (b) How much time does it take to stop?

A car on a circular track accelerates from rest. (a) The car experiences (1) only angular acceleration, (2) only centripetal acceleration, (3) both angular and centripetal accelerations. Why? (b) If the radius of the track is \(0.30 \mathrm{~km}\) and the magnitude of the constant angular acceleration is \(4.5 \times 10^{-3} \mathrm{rad} / \mathrm{s}^{2},\) how long does the car take to make one lap around the track? (c) What is the total (vector) acceleration of the car when it has completed half of a lap?

The blades of a fan running at low speed turn at 250 rpm. When the fan is switched to high speed, the rotation rate increases uniformly to \(350 \mathrm{rpm}\) in \(5.75 \mathrm{~s}\). (a) What is the magnitude of the angular acceleration of the blades? (b) How many revolutions do the blades go through while the fan is accelerating?

In the spin-dry cycle of a modern washing machine, a wet towel with a mass of \(1.50 \mathrm{~kg}\) is "stuck to \({ }^{\prime \prime}\) the inside surface of the perforated (to allow the water out) washing cylinder. To have decent removal of water, damp/wet clothes need to experience a centripetal acceleration of at least \(10 g\). Assuming this value, and that the cylinder has a radius of \(35.0 \mathrm{~cm},\) determine the constant angular acceleration of the towel required if the washing machine takes 2.50 s to achieve its final angular speed.

A simple pendulum of length \(2.00 \mathrm{~m}\) is released from a horizontal position. When it makes an angle of \(30^{\circ}\) from the vertical, determine (a) its angular acceleration, \((\mathrm{b})\) its centripetal acceleration, and \((\mathrm{c})\) the tension in the string. Assume the bob's mass is \(1.50 \mathrm{~kg}\).

A 100 -kg object is taken to a height of \(300 \mathrm{~km}\) above the Earth's surface. (a) What is the object's mass at this height? (b) What is the object's weight at this height?

A man has a mass of \(75 \mathrm{~kg}\) on the Earth's surface. How far above the surface of the Earth would he have to go to "lose" \(10 \%\) of his body weight?

Two objects are attracting each other with a certain gravitational force. (a) If the distance between the objects is halved, the new gravitational force will (1) increase by a factor of 2,(2) increase by a factor of 4,(3) decrease by a factor of 2,(4) decrease by a factor of \(4 .\) Why? (b) If the original force between the two objects is \(0.90 \mathrm{~N},\) and the distance is tripled, what is the new gravitational force between the objects?

An instrument package is projected vertically upward to collect data near the top of the Earth's atmosphere (at an altitude of about \(900 \mathrm{~km}\) ). (a) What initial speed is required at the Earth's surface for the package to reach this height? (b) What percentage of the escape speed is this initial speed?

The asteroid belt that lies between Mars and Jupiter may be the debris of a planet that broke apart or that was not able to form as a result of Jupiter's strong gravitation. An average asteroid has a period of about \(5.0 \mathrm{y}\). Approximately how far from the Sun would this "fifth" planet have been?

Using a development similar to Kepler's law of periods for planets orbiting the Sun, find the required altitude of geosynchronous satellites above the Earth. [Hint: The period of such satellites is the same as that of the Earth.]