Chapter 5

(II) A pilot performs an evasive maneuver by diving vertically at \(310 \mathrm{~m} / \mathrm{s}\). If he can withstand an acceleration of 9.0 's's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?

(III) The position of a particle moving in the \(x y\) plane is given by \(\overrightarrow{\mathbf{r}}=2.0 \cos (3.0 \mathrm{rad} / \mathrm{s} t) \hat{\mathbf{i}}+2.0 \sin (3.0 \mathrm{rad} / \mathrm{s} t) \hat{\mathbf{j}}\) where \(r\) is in meters and \(t\) is in seconds. ( \(a\) ) Show that this represents circular motion of radius \(2.0 \mathrm{~m}\) centered at the origin. (b) Determine the velocity and acceleration vectors as functions of time. ( \(c\) ) Determine the speed and magnitude of the acceleration. \((d)\) Show that \(a=v^{2} / r .(e)\) Show that the acceleration vector always points toward the center of the circle.

(III) A curve of radius \(68 \mathrm{~m}\) is banked for a design speed of \(85 \mathrm{~km} / \mathrm{h}\). If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely make the curve? [Hint: Consider the direction of the friction force when the car goes too slow or too fast.]

(II) A particle starting from rest revolves with uniformly increasing speed in a clockwise circle in the \(x y\) plane. The center of the circle is at the origin of an \(x y\) coordinate system. At \(t=0,\) the particle is at \(x=0.0, y=2.0 \mathrm{~m} .\) At \(t=2.0 \mathrm{~s},\) it has made one-quarter of a revolution and is at \(x=2.0 \mathrm{~m}, y=0.0 .\) Determine \((a)\) its speed at \(t=2.0 \mathrm{~s}\) (b) the average velocity vector, and \((c)\) the average acceleration vector during this interval.

(II) An object moves in a circle of radius \(22 \mathrm{~m}\) with its speed given by \(v=3.6+1.5 t^{2},\) with \(v\) in meters per second and \(t\) in seconds. At \(t=3.0 \mathrm{~s},\) find \((a)\) the tangential acceleration and \((b)\) the radial acceleration.

(III) A particle rotates in a circle of radius \(3.80 \mathrm{~m}\). At a particular instant its acceleration is \(1.15 \mathrm{~m} / \mathrm{s}^{2}\) in a direction that makes an angle of \(38.0^{\circ}\) to its direction of motion. Determine its speed \((a)\) at this moment and \((b) 2.00 \mathrm{~s}\) later, assuming constant tangential acceleration.

(III) An object of mass \(m\) is constrained to move in a circle of radius \(r\). Its tangential acceleration as a function of time is given by \(a_{\tan }=b+c t^{2},\) where \(b\) and \(c\) are constants. If \(v=v_{0}\) at \(t=0,\) determine the tangential and radial components of the force, \(F_{\text {tan }}\) and \(F_{\mathrm{R}}\), acting on the object at any time \(t>0\).

(II) The terminal velocity of a \(3 \times 10^{-5} \mathrm{~kg}\) raindrop is about \(9 \mathrm{~m} / \mathrm{s}\). Assuming a drag force \(F_{\mathrm{D}}=-b v\), determine \((a)\) the value of the constant \(b\) and \((b)\) the time required for such a drop, starting from rest, to reach \(63 \%\) of terminal velocity.

(II) An object moving vertically has \(\overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{v}}_{0}\) at \(t=0\). Determine a formula for its velocity as a function of time assuming a resistive force \(F=-b v\) as well as gravity for two cases: \((a) \overrightarrow{\mathbf{v}}_{0}\) is downward and \((b) \overrightarrow{\mathbf{v}}_{0}\) is upward.

(II) An object moving vertically has \(\vec{\mathbf{v}}=\vec{\mathbf{v}}_{0}\) at \(t=0\) . Determine a formula for its velocity as a function of time assuming a resistive force \(F=-b v\) as well as gravity for two cases: \((a) \vec{\mathbf{v}}_{0}\) is downward and \((b) \vec{\mathbf{v}}_{0}\) is upward.

(III) A bicyclist can coast down a \(7.0^{\circ}\) hill at a steady \(9.5 \mathrm{~km} / \mathrm{h} .\) If the drag force is proportional to the square of the speed \(v,\) so that \(F_{\mathrm{D}}=-c v^{2},\) calculate \((a)\) the value of the constant \(c\) and \((b)\) the average force that must be applied in order to descend the hill at \(25 \mathrm{~km} / \mathrm{h}\). The mass of the cyclist plus bicycle is \(80.0 \mathrm{~kg} .\) Ignore other types of friction.

(III) Two drag forces act on a bicycle and rider: \(F_{\mathrm{D} 1}\) due to rolling resistance, which is essentially velocity independent; and \(F_{\mathrm{D} 2}\) due to air resistance, which is proportional to \(v^{2}\). For a specific bike plus rider of total mass \(78 \mathrm{~kg}\), \(F_{\mathrm{D} 1} \approx 4.0 \mathrm{~N} ;\) and for a speed of \(2.2 \mathrm{~m} / \mathrm{s}, F_{\mathrm{D} 2} \approx 1.0 \mathrm{~N}\) (a) Show that the total drag force is $$ F_{\mathrm{D}}=4.0+0.21 v^{2} $$ where \(v\) is in \(\mathrm{m} / \mathrm{s}\), and \(F_{\mathrm{D}}\) is in \(\mathrm{N}\) and opposes the motion. (b) Determine at what slope angle \(\theta\) the bike and rider can coast downhill at a constant speed of \(8.0 \mathrm{~m} / \mathrm{s}\) s.

(III) Determine a formula for the position and acceleration of a falling object as a function of time if the object starts from rest at \(t=0\) and undergoes a resistive force \(F=-b v,\) as in Example \(5-17\)

(III) A block of mass \(m\) slides along a horizontal surface lubricated with a thick oil which provides a drag force proportional to the square root of velocity: $$ F_{\mathrm{D}}=-b v^{\frac{1}{2}} $$ If \(v=v_{0}\) at \(t=0,\) determine \(v\) and \(x\) as functions of time.

(III) You dive straight down into a pool of water. You hit the water with a speed of \(5.0 \mathrm{~m} / \mathrm{s}\), and your mass is \(75 \mathrm{~kg}\). Assuming a drag force of the form \(F_{\mathrm{D}}=-\left(1.00 \times 10^{4} \mathrm{~kg} / \mathrm{s}\right) v,\) how long does it take you to reach \(2 \%\) of your original speed? (Ignore any effects of buoyancy.)

(III) A motorboat traveling at a speed of \(2.4 \mathrm{~m} / \mathrm{s}\) shuts off its engines at \(t=0 .\) How far does it travel before coming to rest if it is noted that after \(3.0 \mathrm{~s}\) its speed has dropped to half its original value? Assume that the drag force of the water is proportional to \(v\)

A coffee cup on the horizontal dashboard of a car slides forward when the driver decelerates from \(45 \mathrm{~km} / \mathrm{h}\) to rest in \(3.5 \mathrm{~s}\) or less, but not if she decelerates in a longer time. What is the coefficient of static friction between the cup and the dash? Assume the road and the dashboard are level (horizontal).

A \(2.0-\mathrm{kg}\) silverware drawer does not slide readily. The owner gradually pulls with more and more force, and when the applied force reaches \(9.0 \mathrm{~N}\), the drawer suddenly opens, throwing all the utensils to the floor. What is the coefficient of static friction between the drawer and the cabinet?

A 2.0 -kg silverware drawer does not slide readily. The owner gradually pulls with more and more force, and when the applied force reaches \(9.0 \mathrm{N},\) the drawer suddenly opens, throwing all the utensils to the floor. What is the coefficient of static friction between the drawer and the cabinet?

A roller coaster reaches the top of the steepest hill with a speed of \(6.0 \mathrm{~km} / \mathrm{h}\). It then descends the hill, which is at an average angle of \(45^{\circ}\) and is \(45.0 \mathrm{~m}\) long. What will its speed be when it reaches the bottom? Assume \(\mu_{\mathrm{k}}=0.12\).

An \(18.0-\mathrm{kg}\) box is released on a \(37.0^{\circ}\) incline and accelerates down the incline at \(0.220 \mathrm{~m} / \mathrm{s}^{2}\). Find the friction force impeding its motion. How large is the coefficient of friction?

A flat puck (mass \(M\) ) is revolved in a circle on a frictionless air hockey table top, and is held in this orbit by a light cord which is connected to a dangling mass (mass \(m\) ) through a central hole as shown in Fig. \(5-48 .\) Show that the speed of the puck is given by \(v=\sqrt{m g R / M}\)

A motorcyclist is coasting with the engine off at a steady speed of \(20.0 \mathrm{~m} / \mathrm{s}\) but enters a sandy stretch where the coefficient of kinetic friction is \(0.70 .\) Will the cyclist emerge from the sandy stretch without having to start the engine if the sand lasts for \(15 \mathrm{~m} ?\) If so, what will be the speed upon emerging?

A device for training astronauts and jet fighter pilots is designed to rotate the trainee in a horizontal circle of radius \(11.0 \mathrm{~m} .\) If the force felt by the trainee is 7.45 times her own weight, how fast is she rotating? Express your answer in both \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{rev} / \mathrm{s}\)

A \(1250-\mathrm{kg}\) car rounds a curve of radius \(72 \mathrm{~m}\) banked at an angle of \(14^{\circ} .\) If the car is traveling at \(85 \mathrm{~km} / \mathrm{h},\) will a friction force be required? If so, how much and in what direction?

Determine the tangential and centripetal components of the net force exerted on a car (by the ground) when its speed is \(27 \mathrm{~m} / \mathrm{s}\), and it has accelerated to this speed from rest in \(9.0 \mathrm{~s}\) on a curve of radius \(450 \mathrm{~m}\). The car's mass is \(1150 \mathrm{~kg}\).

What is the acceleration experienced by the tip of the 1.5-cm-long sweep second hand on your wrist watch?

A banked curve of radius \(R\) in a new highway is designed so that a car traveling at speed \(v_{0}\) can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, it will slip away from the center of the circle. If the coefficient of static friction increases, it becomes possible for a car to stay on the road while traveling at a speed within a range from \(v_{\min }\) to \(v_{\text { max }}\) . Derive formulas for \(v_{\text { min }}\) and \(v_{\text { max }}\) as functions of \(\mu_{\mathrm{s}}, v_{0},\) and \(R.\)

Earth is not quite an inertial frame. We often make measurements in a reference frame fixed on the Earth, assuming Earth is an inertial reference frame. But the Earth rotates, so this assumption is not quite valid. Show that this assumption is off by 3 parts in 1000 by calculating the acceleration of an object at Earth's equator due to Earth's daily rotation, and compare to \(g=9.80 \mathrm{~m} / \mathrm{s}^{2},\) the acceleration due to gravity.

Consider a train that rounds a curve with a radius of \(570 \mathrm{~m}\) at a speed of \(160 \mathrm{~km} / \mathrm{h}\) (approximately \(100 \mathrm{mi} / \mathrm{h}\) ). ( \(a\) ) Calculate the friction force needed on a train passenger of mass \(75 \mathrm{~kg}\) if the track is not banked and the train does not tilt. (b) Calculate the friction force on the passenger if the train tilts at an angle of \(8.0^{\circ}\) toward the center of the curve.

A car starts rolling down a \(1-\mathrm{in}-4\) hill \((1-\mathrm{in}-4\) means that for each 4 \(\mathrm{m}\) traveled along the road, the elevation change is 1 \(\mathrm{m} ) .\) How fast is it going when it reaches the bottom after traveling 55 \(\mathrm{m}\) ? (a) Ignore friction. (b) Assume an effective coefficient of friction equal to \(0.10 .\)

The sides of a cone make an angle \(\phi\) with the vertical. A small mass \(m\) is placed on the inside of the cone and the cone, with its point down, is revolved at a frequency \(f\) (revolutions per second) about its symmetry axis. If the coefficient of static friction is \(\mu_{\mathrm{s}}\), at what positions on the cone can the mass be placed without sliding on the cone? (Give the maximum and minimum distances, \(r\), from the axis).

A \(72-\mathrm{kg}\) water skier is being accelerated by a ski boat on a flat ("glassy") lake. The coefficient of kinetic friction between the skier's skis and the water surface is \(\mu_{\mathrm{k}}=0.25\) (Fig. \(5-55)\). (a) What is the skier's acceleration if the rope pulling the skier behind the boat applies a horizontal tension force of magnitude \(F_{\mathrm{T}}=240 \mathrm{~N}\) to the skier \(\left(\theta=0^{\circ}\right) ?\) (b) What is the skier's horizontal acceleration if the rope pulling the skier exerts a force of \(F_{\mathrm{T}}=240 \mathrm{~N}\) on the skier at an upward angle \(\theta=12^{\circ} ?\) (c) Explain why the skier's acceleration in part ( \(b\) ) is greater than that in part ( \(a\) ).

A 72 -kg water skier is being accelerated by a ski boat on a flat \((\) "glassy") lake. The coefficient of kinetic friction between the skier's skis and the water surface is \(\mu_{k}=0.25\) (Fig. 55 ). (a) What is the skier's acceleration if the rope pulling the skier behind the boat applies a horizontal tension force of magnitude \(F_{T}=240 \mathrm{N}\) to the skier \(\left(\theta=0^{\circ}\right) ?\) (b) What is the skier's horizontal acceleration if the rope pulling the skier exerts a force of \(F_{T}=240 \mathrm{N}\) on the skier at an upward angle \(\theta=12^{\circ} ?\) (c) Explain why the skier's acceleration in part \((b)\) is greater than that in part \((a)\) .

A ball of mass \(m=1.0 \mathrm{~kg}\) at the end of a thin cord of length \(r=0.80 \mathrm{~m}\) revolves in a vertical circle about point \(\mathrm{O},\) as shown in Fig. \(5-56 .\) During the time we observe it, the only forces acting on the ball are gravity and the tension in the cord. The motion is circular but not uniform because of the force of gravity. The ball increases in speed as it descends and decelerates as it rises on the other side of the circle. At the moment the cord makes an angle \(\theta=30^{\circ}\) below the horizontal, the ball's speed is \(6.0 \mathrm{~m} / \mathrm{s}\). At this point, determine the tangential acceleration, the radial acceleration, and the tension in the cord, \(F_{\mathrm{T}}\). Take \(\theta\) increasing downward as shown.

A car drives at a constant speed around a banked circular track with a diameter of 127 \(\mathrm{m}\) . The motion of the car can be described in a coordinate system with its origin at the center of the circle. At a particular instant the car's accel- eration in the horizontal plane is given by $$\vec{\mathbf{a}}=(-15.7 \hat{\mathbf{i}}-23.2 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2}$$ (a) What is the car's speed? (b) Where \((x\) and \(y)\) is the car at this instant?

(III) The force of air resistance (drag force) on a rapidly falling body such as a skydiver has the form \(F_{\mathrm{D}}=-k v^{2},\) so that Newton's second law applied to such an object is $$ m \frac{d v}{d t}=m g-k v^{2} $$ where the downward direction is taken to be positive. (a) Use numerical integration [Section \(2-9]\) to estimate (within \(2 \%\) ) the position, speed, and acceleraton, from \(t=0\) up to \(t=15.0 \mathrm{~s},\) for a \(75-\mathrm{kg}\) skydiver who starts from rest, assuming \(k=0.22 \mathrm{~kg} / \mathrm{m}\) (b) Show that the diver eventually reaches a steady speed, the terminal speed, and explain why this happens. (c) How long does it take for the skydiver to reach \(99.5 \%\) of the terminal speed?

(III) The coefficient of kinetic friction \(\mu_{\mathrm{k}}\) between two surfaces is not strictly independent of the velocity of the object. A possible expression for \(\mu_{\mathrm{k}}\) for wood on wood is $$\mu_{\mathrm{k}}=\frac{0.20}{\left(1+0.0020 v^{2}\right)^{2}},$$ where \(v\) is in \(\mathrm{m} / \mathrm{s} .\) A wooden block of mass 8.0 \(\mathrm{kg}\) is at rest on a wooden floor, and a constant horizontal force of 41 \(\mathrm{N}\) acts on the block. Use numerical integration to determine and graph \((a)\) the speed of the block, and \((b)\) its position, as a function of time from 0 to 5.0 \(\mathrm{s}\) (c) Determine the percent difference for the speed and position at 5.0 \(\mathrm{s}\) if \(\mu_{\mathrm{k}}\) is constant and equal to \(0.20 .\)

(III) Assume a net force \(F=-m g-k v^{2}\) acts during the upward vertical motion of a \(250-\mathrm{kg}\) rocket, starting at the moment \((t=0)\) when the fuel has burned out and the rocket has an upward speed of \(120 \mathrm{~m} / \mathrm{s}\). Let \(k=0.65 \mathrm{~kg} / \mathrm{m}\). Estimate \(v\) and \(y\) at 1.0 -s intervals for the upward motion only, and estimate the maximum height reached. Compare to free-flight conditions without air resistance \((k=0)\).