MMP: Ch. 8 and 9

Is the maximum possible bite force applied with your molars, in the back of your mouth, larger or smaller than with your incisors, at the front of your mouth? Choose the correct explanation.

Larger. The moment arm between molars and the axis around which the jaw rotates is smaller compared to incisors, which makes the force larger in order to develop the same torque.

A steel wire of length 1.95 m with circular cross section must stretch no more than 0.300 cm when a tensile (stretching) force of 370 N is applied to each end of the wire. What minimum diameter dmin is required for the wire?

1.24 mm

When his leg is stretched out before him, what is the torque exerted by the weight about his knee, 50 cm away from the weight? Use g = 10 m/s^2

20 N

A 63 kg diver stands at the end of a 29 kg springboard, as shown in the figure. The board is attached to a hinge at the left end but simply rests on the right support. What is the magnitude of the vertical force exerted by the hinge on the board?

620 N

The spring constant can be adjusted with Spring Constant slider bar ("larger" means a greater spring constant, or stiffer spring). How does the frequency of oscillation depend on the spring constant?

The frequency increases as the spring constant increases. The frequency of oscillation depends on the square root of the ratio of the spring constant to mass: f= √/2πf=(k/m)/2π, where f is the frequency. A stiffer spring constant causes the frequency to increase. Sports cars use stiff springs for their suspension, whereas large plush Cadillacs use soft springs.

When the ball hangs from the ligament, it stretches a certain amount. If the ball is then pulled downward, the ligament experiences a greater stretch. When the ball is subsequently released, what is the direction of the net force on the ball?

The net force is upward

Should you choose to throw a rubber ball, which will bounce off the pin, or a beanbag, which will strike the pin and not bounce? Assume the ball and beanbag have equal size and weight.

rubber ball

Model rocket engines are rated by the impulse that they deliver when they fire. A particular engine is rated to deliver an impulse of 3.5 kg⋅m/s. The engine powers a 140g rocket, including the mass of the engine. What is the final speed of the rocket once the engine has fired? (Ignore the change in mass as the engine fires and ignore the weight force during the short duration of the engine firing.)

25 m/s

A vendor hangs an 8.2 kg sign in front of his shop with a cable held away from the building by a lightweight pole. The pole is free to pivot about the end where it touches the wall, as shown in (Figure 1). What is the tension in the cable?

160 N

Experiments using "optical tweezers" measure the elasticity of individual DNA molecules. For small enough changes in length, the elasticity has the same form as that of a spring. A DNA molecule is anchored at one end, then a force of 1.5nN (1.5×10−9N)(1.5×10−9N) pulls on the other end, causing the molecule to stretch by 5.0nm (5.0×10−9m)(5.0×10−9m). What is the spring constant of that DNA molecule?

0.3 N/m

A trap-jaw ant snaps its mandibles shut at very high speed, a good trait for catching small prey. But an ant can also slam its mandibles into the ground; the resulting force can launch the ant into the air for a quick escape. A 12 mg ant hits the ground with an average force of 47 mN for a time of 0.13 ms; these are all typical values. At what speed does it leave the ground?

0.51 m/s

A 57 g tennis ball is served at 45 m/s . If the ball started from rest, what impulse was applied to the ball by the racket?

P = 2.565 kgm/s

DNA molecules are typically folded tightly. Stretching a strand of DNA means straightening it, and the molecules resist this straightening. Investigators can attach beads to the ends of a strand of DNA and, using "optical tweezers," measure the force required to produce a certain extension. Data for the stretch of a 3500 base pair strand of DNA approximately follow the line in the graph in (Figure 1). What is the spring constant for this strand of DNA?

k = 2.5×10^−7 N/m

The normal force of the ground on the foot can reach three times a runner's body weight when the foot strikes the pavement. By what amount does the 52-cm-long femur of an 75 kg runner compress at this moment? The cross-section area of the bone of the femur can be taken as 5.2×10−4m^2 and its Young's modulus is 1.6×10^10N/m^2.

1.36 X 10^-4 m

The parking brake on a 1000 kg Cadillac has failed, and it is rolling slowly, at 4 mph , toward a group of small children. Seeing the situation, you realize you have just enough time to drive your 2000 kg Volkswagen head-on into the Cadillac and save the children. With what speed should you impact the Cadillac to bring it to a halt?

2 mph

Erica (39 kg ) and Danny (46 kg ) are bouncing on a trampoline. Just as Erica reaches the high point of her bounce, Danny is moving upward past her at 4.4 m/s . At that instant he grabs hold of her. What is their speed just after he grabs her?

2.4 m/s

Select the stopwatch, and time how long it takes for a weight to oscillate back and forth 10 times (a single oscillation is represented by a return to a weight's original position). The period of oscillation T is this time divided by 10. The frequency of oscillation is 1/T. How does the frequency of oscillation depend on the value of the mass?

The frequency decreases as the mass increases. A larger mass results in a lower frequency and a longer period of oscillation.

A firecracker in a coconut blows the coconut into three pieces. Two pieces of equal mass fly off south and west, perpendicular to each other, at 22 m/s . The third piece has twice the mass as the other two. What is the speed of the third piece? What is the direction of the third piece?

15.55 m/s 45 degrees north of east

A bike chain can support a tension of no more than 9800 N. The pedal connects to a crank 17 cm from the axle, and the gear pulling the chain has a 9.1 cm radius. When riding at a constant speed, with the crank and pedal horizontal, as in (Figure 1), what is the maximum force that can be applied to the pedal before the chain breaks?

5245.88 N

When the lever is pulled, 2 kg of carbon dioxide is ejected at a speed of 60 m/s. The remaining mass of the person, chair, and cylinder is 80 kg. After the ejection, how fast will the chair be moving?

1.5 m/s

A hippo's body is 4.0 mm long with front and rear feet located as in (Figure 1). The hippo carries 60%% of its weight on its front feet. How far from its tail is the hippo's center of gravity?

1.58 m

While unrealistic, we will examine the forces on a leg when one falls from a height by approximating the leg as a uniform cylinder of bone with a diameter of 2.3 cm and ignoring any shear forces. Human bone can be compressed with approximately 1.7 × 1088 N/m^2 before breaking. A man with a mass of 70 kg falls from a height of 6 m. Assume his acceleration once he hits the ground is constant. For these calculations, g = 10 m/s^2 With how much force can the "leg" be compressed before breaking? What is his speed just before he hits the ground? If he lands "stiff legged" and his shoes only compress 1 cmcm, what is the magnitude of the average force he experiences as he slows to a rest? If he bends his legs as he lands, he can increase the distance over which he slows down to 50 cmcm. What would be the average force he experiences in this scenario? Dyne is also a unit of force and 1 Dyn = 10^−5N. What is the maximum a bone can be compressed in Dyn/cm^2? Which of the following is the reason that we would recommend that the man bend his legs while landing from such a fall?

11 m/s 7.1 X 10^4 N 4.2 X 10^5 N 8.4 X 10^3 N 1.7 X 10^9 Dyn/cm^2 Bending his legs increases the time over which the ground applies force, thus decreasing the force applied by the ground.

If you tethered a space station to the earth by a long cable, you could get to space in an elevator that rides up the cable--much simpler and cheaper than riding to space on a rocket. There's one big problem, however: There is no way to create a cable that is long enough. The cable would need to reach 36,000 km upward, to the height where a satellite orbits at the same speed as the earth rotates; a cable this long made of ordinary materials couldn't even support its own weight. Consider a steel cable suspended from a point high above the earth. The stress in the cable is highest at the top; it must support the weight of cable below it. What is the greatest length the cable could have without failing? For the purposes of this problem, you can ignore the variation in gravity near the surface of the earth. Hint: The mass of the cable is the volume of the cable multiplied by the density. The density of steel is 7900 kg/m^3

12.9 km

A diver leaves the platform with her body straight. Her body is in a relatively slow rotation, with an angular speed of 4.0 rad/s. She then tucks into a pike position, with her body essentially folded in half. We can use a simple model to understand what happens next. First, model her 50 kg, 1.8 m body as uniform. Next, assume that when she goes into a pike position, she really does fold her body exactly in half. In terms of this model, what is her initial moment of inertia? In terms of this model, what is her moment of inertia in the pike position? In terms of this model, what is her angular speed in the pike position? In terms of this model, how many rotations does she complete in the 1.3 s that she holds the pike position?

14kg⋅m^2 3.4 kg⋅m^2 16 rad/s 3.3 rotations

A standard four-drawer filing cabinet is 52 inches high and 15 inches wide. If it is evenly loaded, the center of gravity is at the center of the cabinet. A worker is tilting a filing cabinet to the side to clean under it. To what angle can he tilt the cabinet before it tips over?

16 Degrees

The main muscles that hold your head upright attach to your spine in back of the point where your head pivots on your neck. (Figure 1) shows typical numbers for the distance from the pivot to the muscle attachment point and the distance from the pivot to the center of gravity of the head. The muscles pull down to keep your head upright. If the muscle relaxes--if, for instance, you doze in one of your classes besides Physics--your head tips forward. In the questions that follow, assume that your head has a mass of 4.8 kg, and that you maintain the relative angle between your head and your spine. With the head held level, as in (Figure 1), what muscle force is needed to keep a 4.8 kg head upright? If you tip your body forward so that your spine is level with the ground, what muscle force is needed to keep your head in the same orientation relative to the spine? If you tip your body backward, you will reach a point where no muscle force is needed to keep your head upright. For the distances given in (Figure 1), at what angle does this balance occur?

39.5 N 94 N 22.8 degrees

The nuchal ligament in a horse supports the weight of the horse's head. This ligament is much more elastic than a typical ligament, stretching from 15%% to 45%% longer than its resting length as a horse's head moves up and down while it runs. This stretch of the ligament stores energy, making locomotion more efficient. Measurements on a segment of ligament show a linear stress-versus-strain relationship until the strain approaches 0.80. Suppose the ligament has a circular cross section. For a certain ligament, an investigator measures the restoring force at a strain of 0.40. If the ligament is replaced with one that has twice the diameter, by what factor does the restoring force increase?

4

The combined mass of the wheelbarrow and the load is 140kg, with a center of gravity at dd = 0.35m behind the axle. The woman supports the wheelbarrow at the handles, 1.1 m behind the axle. What is the force required to support the wheelbarrow? What fraction of the weight of the wheelbarrow and the load does this force represent?

437 N 0.318

In the hammer throw, an athlete spins a heavy mass in a circle at the end of a cable before releasing it for distance as shown in (Figure 1). For male athletes, the "hammer" is a mass of 7.3 kg at the end of a 1.2 mm cable, which is typically a 3.0-mm-diameter steel cable. A world-class thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the cable What is the tension in the cable? Neglect the gravity. How much does the cable stretch? Young modulus for steel is 20×10^10N/m^2

5100 N 4.3 mm

At the county fair, Chris throws a 0.15 kg baseball at a 2.5 kg wooden milk bottle, hoping to knock it off its stand and win a prize. The ball bounces straight back at 10 % of its incoming speed, knocking the bottle straight forward. What is the bottle's speed, as a percentage of the ball's incoming speed?

6.6 %

A child is sliding on a sled at 1.3 m/s to the right. You stop the sled by pushing on it for 0.80 s in a direction opposite to its motion. If the mass of the child and sled is 39 kg , what is the magnitude of the average force you need to apply to stop the sled? Use the concepts of impulse and momentum.

63 N

In principle, when you fire a rifle, the recoil should push you backward. How big a push will it give? Let's find out by doing a calculation in a very artificial situation. Suppose a man standing on frictionless ice fires a rifle horizontally. The mass of the man together with the rifle is 70 kg, and the mass of the bullet is 10 g. If the bullet leaves the muzzle at a speed of 500 m/s, what is the final speed of the man?

7.1×10^−2 m/s

A typical raindrop is much more massive than a mosquito and falling much faster than a mosquito flies. How does a mosquito survive the impact? Recent research has found that the collision of a falling raindrop with a mosquito is a perfectly inelastic collision. That is, the mosquito is "swept up" by the raindrop and ends up traveling along with the raindrop. Once the relative speed between the mosquito and the raindrop is zero, the mosquito is able to detach itself from the drop and fly away. A hovering mosquito is hit by a raindrop that is 40 times as massive and falling at 8.9 m/s , a typical raindrop speed. How fast is the raindrop, with the attached mosquito, falling immediately afterward if the collision is perfectly inelastic? Because a raindrop is "soft" and deformable, the collision duration is a relatively long 8.0 m/s. What is the mosquito's average acceleration, in g's, during the collision? The peak acceleration is roughly twice the value you found, but the mosquito's rigid exoskeleton allows it to survive accelerations of this magnitude. In contrast, humans cannot survive an acceleration of more than about 10 g.

8.68 m/s 110 g

The amplitude of oscillation is the maximum distance between the oscillating weight and the equilibrium position. Determine the frequency of oscillation for several different amplitudes by pulling the weight down different amounts while still keeping the simulation within the top and bottom boundaries. How does the frequency depend on the amplitude of oscillation?

The frequency is independent of the amplitude. Even though the weight has to travel farther in each oscillation if the amplitude is greater, the spring on average exerts a stronger force, causing a greater acceleration and a greater average speed. The effects of the longer distance and faster speed cancel out so that the period of oscillation doesn't change!

An 80-kg quarterback jumps straight up in the air right before throwing a 0.43-kg football horizontally at 15 m/s . How fast will he be moving backward just after releasing the ball? Which interaction will you analyze in this problem? Which of the following best describes why you can analyze this event using conservation of momentum? How fast, (vQx)f, will the quarterback be moving backward just after releasing the ball? Suppose that the quarterback takes 0.30 s to return to the ground after throwing the ball. How far d will he move horizontally, assuming his speed is constant?

The quarterback throwing the ball. The throwing action is quick enough that external forces may be ignored. (vQx)f = 8.1×10^-2m/s 2.4 X 10^-2 m

Take a spring and cut it in half to make two springs. How does the spring constant of the smaller springs relate to that of the original spring? Choose the correct explanation.

The spring constant of each half will be twice the spring constant of the original long spring since it will stretch only half as much under the same tension.

Place the single weight with a known mass on the spring and release it. Eventually, the weight will come to rest at an equilibrium position, with the spring somewhat stretched compared to its original (unweighted) length. At this point, the upward force of the spring balances the force of gravity on the weight. With the weight in its equilibrium position, how does the amount the spring is stretched depend on the value of the weight's mass?

The spring stretches more for a larger mass Since the force of gravity on the weight increases as the value of the mass increases, the upward force of the spring must increase for the two forces to balance (and the weight to therefore be in equilibrium). The force the spring exerts on the weight increases as the spring is stretched more from its unweighted length.

When is the kinetic energy of the mass a maximum?

When the spring is at its unweighted length (when it isn't stretched or compressed) The total energy of the system is equal to the kinetic energy of the weight (since the spring has negligible mass) plus the elastic potential energy. Since this total energy is conserved, the kinetic energy is a maximum when the elastic potential energy is a minimum, which occurs when the spring is at its unweighted length.

Dynamic climbing ropes are designed to be quite pliant, allowing a falling climber to slow down over a long distance. The graph in (Figure 1) shows force-versus-strain data for an 11- mm -diameter climbing rope. What is the Young's modulus for this rope?

Y = 3.65×10^8 N/m^2

Find the tangent of the angle θ Suppose that after the collision, tan⁡θ=1; in other words, θ is 45 degrees. Which quantities then must have been equal before the collision?

tan⁡(θ) = m2v2m1v1 The magnitudes of the momenta of the cars

The Achilles tendon connects the muscles in your calf to the back of your foot. When you are sprinting, your Achilles tendon alternately stretches, as you bring your weight down onto your forward foot, and contracts to push you off the ground. A 70 kg runner has an Achilles tendon that is 15 cm long with a typical 1.1×10^−4m^2 area. By how much will the runner's Achilles tendon stretch if the maximum force on it is 8.0 times his weight, a typical value while running? To what fraction of the tendon's length does this correspond?

ΔL = 5.0×10^−3 m ΔL/L = 3.3×10^−2

You're carrying a 3.8-m-long, 20 kg pole to a construction site when you decide to stop for a rest. You place one end of the pole on a fence post and hold the other end of the pole 35 cm from its tip. How much force must you exert to keep the pole motionless in a horizontal position?

107.94 N

Hold your upper arm vertical and your lower arm horizontal with your hand palm-down on a table, as shown in (Figure 1). If you now push down on the table, you'll feel that your triceps muscle has contracted and is trying to pivot your lower arm about the elbow joint. If a person with the arm dimensions shown pushes down hard with a 90 NN force, what force must the triceps muscle provide? Ignore the mass of the arm and hand in your calculation.

1125 N

When the ligament is stretched, it exerts a restoring force. A stretch of 4 cm produces a restoring force of 60 N. What is the restoring force for a 2 cm stretch?

30 N

Peregrine falcons frequently grab prey birds from the air. Sometimes they strike at high enough speeds that the force of the impact disables prey birds. A 480 g peregrine falcon high in the sky spies a 240 g pigeon some distance below. The falcon slows to a near stop, then goes into a dive--called a stoop--and picks up speed as she falls. The falcon reaches a vertical speed of 45 m/s before striking the pigeon, which we can assume is stationary. The falcon strikes the pigeon and grabs it in her talons. The collision between the birds lasts 0.015 s. What is the final speed of the falcon and pigeon? What is the average force on the pigeon during the impact?

30 m/s 480 N

A 10-cm-long spring is attached to the ceiling. When a 2.0 kg mass is hung from it, the spring stretches to a length of 15 cm. What is the spring constant k? How long is the spring when a 3.9 kg mass is suspended from it?

400Nm 19.75 cm

When you lift an object by moving only your forearm, the main lifting muscle in your arm is the biceps. Suppose the mass of a forearm is 1.50 kg . If the biceps is connected to the forearm a distance dbiceps = 3.50 cm from the elbow, how much force Fbiceps must the biceps exert to hold a 500 gg ball at the end of the forearm at distance dball = 32.0 cm from the elbow, with the forearm parallel to the floor? How much force Felbow must the elbow exert? The forearm should be modeled as What is the horizontal distance dforearm between the elbow and the point where the weight of the forearm acts? Determine the sign (++ or −−) of the torque about the elbow caused by the biceps, τbiceps, the sign of the weight of the forearm, τforearm, and the sign of the weight of the ball, τball. Write an expression for ∑Fy. Take upward to be the positive y direction. Write an expression for ∑τ about the elbow. What is the magnitude of the force exerted by the biceps Fbiceps? What is the magnitude of the force exerted by the elbow Felbow? Although it is uncommon, orthopedic surgeons have seen patients who have torn their biceps tendon when forces of greater than 390 NN have been exerted on the tendon with the arm bent at the elbow so that the forearm is parallel to the ground. What is the minimum mass of a ball Mball such a patient might have held to cause the biceps tendon to tear?

A rigid rod in equilibrium 0.160 m +,-,- = −Felbow +Fbiceps − mforearmg −mballg ∑τ = dbiceps X Fbiceps − mforearmg (dball/2) − mball X g X dball Fbiceps, Felbow = 112, 92.4 N,N Mball = 3.6 kg

We've seen that squid can escape from predators by ejecting water. Some squid do this at the surface of the ocean, thus launching themselves into the air-a particularly effective escape strategy. Suppose a 36 kg squid (not including water) at rest at the surface of the water brings in and quickly ejects 3.0 kg of water to achieve a takeoff speed of 3.5 m/s; these are typical numbers. At what speed does the squid eject the water? If we ignore lift and drag forces as the squid flies through the air, what is the maximum horizontal range that the squid can achieve before splashing down?

42 m/s 1.3 m

Select Earth in the menu box so that there is now a force of gravity. Now the total energy of the weight/spring/Earth system is the sum of the kinetic energy, the elastic potential energy, and the gravitational potential energy. When is the kinetic energy a maximum? (It may help to watch the simulation in slow motion).

When the mass is at the equilibrium position The total potential energy (the gravitational potential energy plus elastic potential energy) is a minimum at the equilibrium position, even though there is some elastic potential energy when the weight is at this location. The kinetic energy is always a maximum when the total potential energy is a minimum (since the total energy is conserved).

A double-decker London bus (Figure 1) might be in danger of rolling over in a highway accident, but at the low speeds of its urban environment, it's plenty stable. The track width is 2.05 m. With no passengers, the height of the center of gravity is 1.45 m, rising to 1.73 m when the bus is loaded to capacity. What is the critical angle for the unloaded bus? What is the critical angle for the loaded bus?

35.3 degrees 30.6 degrees

In the past, asteroids striking the earth have produced disastrous results. If we discovered an asteroid on a collision course with the earth, we could, in principle, deflect it and avoid an impact by focusing a laser on the surface. Intense surface heating from the laser could cause surface material to be ejected into space at high speed. How would this deflect the asteroid?

The material ejected from the surface of asteroid would have a significant momentum. Since asteroid and all its material is , the ejection would cause change in momentum of the asteroid, according to the . The ejected material is analogous to , and the asteroid is analogous to . an isolated system oppositely directly law of conservation of momentum Gases expelled from a rocket the rocket

A water pipe in a building delivers 1000 liters (with mass 1000 kg) of water per second. The water is moving through the pipe at 2.2 m/s. The pipe has a 90∘∘ bend, and the pipe will require a supporting structure, called a thrust block, at the bend, as in (Figure 1) . We can use the ideas of momentum and impulse to understand why. Each second, 1000 kg of water moving at vxvx = 2.2 m/s changes direction to move at vy = 2.2 m/s. What is the magnitude of the change in momentum of the 1000 kg of water? What is the direction of the change in momentum of water? What is the magnitude of the necessary impulse? What is the direction of the necessary impulse? This impulse takes place over 1.0 ss. What is the magnitude of the necessary force? What is the direction of the necessary force?

3100 kg⋅m/s At angle of 45∘∘ from both segments of the pipe. 3100 N At angle of 45∘∘ from both segments of the pipe. 3100 N At angle of 45∘∘ from both segments of the pipe.

An 85 kg man stands in a very strong wind moving at 16 m/s at torso height. As you know, he will need to lean in to the wind, and we can model the situation to see why. Assume that the man has a mass of 85 kg, with a center of gravity 1.0 m above the ground. The action of the wind on his torso, which we approximate as a cylinder 50 cm wide and 90 cm long centered 1.2 m above the ground, produces a force that tries to tip him over backward. (Drag coefficient for the cylinder in this position is CDCD = 1.1.) To keep from falling over, he must lean forward. What is the magnitude of the torque provided by the wind force? Take the pivot point at his feet. Assume that he is standing vertically. Assume that the air is at standard temperature and pressure. At what angle to the vertical must the man lean to provide a gravitational torque that is equal to this torque due to the wind force?

91 Nm 6.3 degrees

You should see an Energy Graph to the left. If not, make sure you are in the correct simulation. Set the Gravity to "None" (g = 0), which simulates what happens without any gravitational forces (and consequently removes gravitational potential energy from the energetics). Adjust the damping slider to "None" (this prevents any energy dissipation). Place the weight on the spring and set its mass value to 125 g. Set the spring constant to "Large." Next, stretch the spring and release the mass. The amplitude of the stretch should be large enough to get good resoluton on the Energy Graph, but not so much that the spring overshoots at the top. Observe how the kinetic energy and elastic potential energy vary with time. Note: you can slow down or stop time using the controls in the lower right of the simulation window. When is the elastic potential energy of the spring a maximum?

Both when the spring is most compressed and when the spring is most stretched

A passenger railroad car has a total of 8 wheels. Springs on each wheel compress--slightly--when the car is loaded. Ratings for the car give stiffness per wheel (the spring constant, treating the entire spring assembly as a single spring) as 2.8×107N/m^2. When 30 passengers, each with average mass of 80 kg, board the car, how much does the car move down on its spring suspension? Assume that each wheel supports 1//8 the weight of the car.

Δy = 1.1×10^−4 m

As part of a safety investigation, two 1800 kg cars traveling at 24 m/s are crashed into different barriers. Find the average force exerted on the car that hits a line of water barrels and takes 1.8 s to stop. Find the average force exerted on the car that hits a concrete barrier and takes 0.14 s to stop.

2.4 X10^4 N 3.1×10^5 N

The 630 gg bar is rotating as shown in What is the angular momentum of the bar about the axle?

2.6 kg⋅m^2/s