Problems & Exercises

(a) When rebuilding her car’s engine, a physics major must exert 300 N of force to insert a dry steel piston into a steel cylinder. What is the magnitude of the normal force between the piston and cylinder? (b) What is the magnitude of the force would she have to exert if the steel parts were oiled?

(a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous exercise it is possible to exert forces to the joints that are easily ten times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pain.

(a) If half of the weight of a small $1\text{.}\text{00}\times {\text{10}}^3\text{kg}$utility truck is supported by its two drive wheels, what is the magnitude of the maximum acceleration it can achieve on dry concrete? (b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate? (c) Solve both problems assuming the truck has four-wheel drive.

Consider the 65.0-kg ice skater being pushed by two others shown in Figure 5.19. (a) Find the direction and magnitude of ${\mathbf{F}}_{\text{tot}}$, the total force exerted on her by the others, given that the magnitudes $F_1$ and $F_2$ are 26.4 N and 18.6 N, respectively. (b) What is her initial acceleration if she is initially stationary and wearing steel-bladed skates that point in the direction of ${\mathbf{F}}_{\text{tot}}$? (c) What is her acceleration assuming she is already moving in the direction of ${\mathbf{F}}_{\text{tot}}$? (Remember that friction always acts in the direction opposite that of motion or attempted motion between surfaces in contact.)

Show that the acceleration of any object down a frictionless incline that makes an angle $\theta$ with the horizontal is $a=g\text{sin}\theta$. (Note that this acceleration is independent of mass.)

Show that the acceleration of any object down an incline where friction behaves simply (that is, where $f_{\text{k}}={\mu }_{\text{k}}N$) is $a=g(\text{sin}\theta -{\mu }_{\text{k}}\text{cos}\theta ).$ Note that the acceleration is independent of mass and reduces to the expression found in the previous problem when friction becomes negligibly small $({\mu }_{\text{k}}=0).$

(a) Calculate the acceleration of a skier heading down a $\text{10}\text{.}\mathrm{0º}$ slope, assuming the coefficient of friction for waxed wood on wet snow. (b) Find the angle of the slope down which this skier could coast at a constant velocity. You can neglect air resistance in both parts, and you will find the result of Exercise 5.9 to be useful. Explicitly show how you follow the steps in the Problem-Solving Strategies.

If an object is to rest on an incline without slipping, then friction must equal the component of the weight of the object parallel to the incline. This requires greater and greater friction for steeper slopes. Show that the maximum angle of an incline above the horizontal for which an object will not slide down is $\theta ={\text{tan}}^{\text{–1}}{\text{μ}}_{\text{s}}$. You may use the result of the previous problem. Assume that $a=\text{0}$ and that static friction has reached its maximum value.

Calculate the maximum deceleration of a car that is heading down a $\mathrm{6º}$ slope (one that makes an angle of $\mathrm{6º}$ with the horizontal) under the following road conditions. You may assume that the weight of the car is evenly distributed on all four tires and that the coefficient of static friction is involved—that is, the tires are not allowed to slip during the deceleration. (Ignore rolling.) Calculate for a car: (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that ${\mu }_{\mathrm{s}}=0\text{.}\text{100}$, the same as for shoes on ice.

Repeat Exercise 5.14 for a car with four-wheel drive.

Consider the 52.0-kg mountain climber in Figure 5.20. (a) Find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock face to remain stationary. Assume that the force is exerted parallel to her legs. Also, assume negligible force exerted by her arms. (b) What is the minimum coefficient of friction between her shoes and the cliff?

Repeat Exercise 5.18 with the contestant pulling the block of ice with a rope over his shoulder at the same angle above the horizontal as shown in Figure 5.21(b).

A 60-kg and a 90-kg skydiver jump from an airplane at an altitude of 6000 m, both falling in a headfirst position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits.

To maintain a constant speed, the force provided by a car’s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the magnitudes of drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Drag area is ${\text{0.70 m}}^2$) (b) What is the magnitude of drag force at 70 km/h and 100 km/h for a Hummer H2? (Drag area is $2\text{.}{\text{44 m}}^2$) Assume all values are accurate to three significant digits.

Calculate the speed a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be $1\text{.}\text{00}\times {\text{10}}^3{\text{kg/m}}^3$, and the cross-section area facing the fluid to be ${\mathrm{\pi r}}^2$.

Find the terminal velocity of a spherical bacterium (diameter $\mathrm{2.00\; \mu m}$) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be $1\text{.}\text{10}\times {\text{10}}^3{\text{kg/m}}^3$.

During a wrestling match, a 150 kg wrestler briefly stands on one hand during a maneuver designed to perplex her adversary. By how much does the upper arm bone shorten in length? The bone can be represented by a uniform rod 38.0 cm in length and 2.10 cm in radius.

TV broadcast antennas are the tallest artificial structures on Earth. In 1987, a 72.0-kg physicist placed himself and 400 kg of equipment at the top of one 610-m high antenna to perform gravity experiments. By how much was the antenna compressed, if we consider it to be equivalent to a steel cylinder 0.150 m in radius?

A 20.0-m tall hollow aluminum flagpole is equivalent in stiffness to a solid cylinder 4.00 cm in diameter. A strong wind bends the pole much as a horizontal force of 900 N exerted at the top would. How far to the side does the top of the pole flex?

Calculate the force a piano tuner applies to stretch a steel piano wire 8.00 mm, if the wire is originally 0.850 mm in diameter and 1.35 m long.

A disk between vertebrae in the spine is subjected to a shearing force of 600 N. Find its shear deformation, taking it to have the shear modulus of $1\times {\text{10}}^9\text{N}/{\text{m}}^2$ . The disk is equivalent to a solid cylinder 0.700 cm high and 4.00 cm in diameter.

To consider the effect of wires hung on poles, we take data from Example 4.8, in which tensions in wires supporting a traffic light were calculated. The left wire made an angle $\text{30.0º}$ below the horizontal with the top of its pole and carried a tension of 108 N. The 12.0 m tall hollow aluminum pole is equivalent in stiffness to a 4.50 cm diameter solid cylinder. (a) How far is it bent to the side? (b) By how much is it compressed?

(a) When water freezes, its volume increases by 9.05% (that is, $\mathrm{\Delta }V/V_0=9\text{.}\text{05}\times {\text{10}}^{-2}$ ). What force per unit area is water capable of exerting on a container when it freezes? (It is acceptable to use the bulk modulus of water in this problem.) (b) Is it surprising that such forces can fracture engine blocks, boulders, and the like?

The pole in Figure 5.22 is at a $\text{90.0º}$ bend in a power line and is therefore subjected to more shear force than poles in straight parts of the line. The tension in each line is $4\text{.}\text{00}\times {\text{10}}^4\mathrm{N}$, at the angles shown. The pole is 15.0 m tall, has an 18.0 cm diameter, and can be considered to have half the stiffness of hardwood. (a) Calculate the compression of the pole. (b) Find how much it bends and in what direction. (c) Find the tension in a guy wire used to keep the pole straight if it is attached to the top of the pole at an angle of $\text{30.0º}$ with the vertical. (Clearly, the guy wire must be in the opposite direction of the bend.)