Solved: In the game of paintball, players use guns powered

In what ways is the word work as used in everyday language the same as defined in physics? In what ways is it different? Give examples of both.

A woman swimming upstream is not moving with respect to the shore. Is she doing any work? If she stops swimming and merely floats, is work done on her?

Can a centripetal force ever do work on an object? Explain.

Why is it tiring to push hard against a solid wall even though you are doing no work?

Does the scalar product of two vectors depend on the choice of coordinate system?

Can a dot product ever be negative? If yes, under what conditions?

If A C = B C, is it necessarily true that A = B?

Does the dot product of two vectors have direction as well as magnitude?

Can the normal force on an object ever do work? Explain.

You have two springs that are identical except that spring 1 is stiffer than spring 2 (k\ > /c2). On which spring is more work done: (a) if they are stretched using the same force; (ib) if they are stretched the same distance?

If the speed of a particle triples, by what factor does its kinetic energy increase

In Example 7-10, it was stated that the block separates from the compressed spring when the spring reached its equilibrium length (x = 0). Explain why separation doesnt take place before (or after) this point.

Two bullets are fired at the same time with the same kinetic energy. If one bullet has twice the mass of the other, which has the greater speed and by what factor? Which can do the most work?

Does the net work done on a particle depend on the choice of reference frame? How does this affect the work-energy principle?

A hand exerts a constant horizontal force on a block that is free to slide on a frictionless surface (Fig. 7-19). The block starts from rest at point A, and by the time it has traveled a distance d to point B it is traveling with speed vB. When the block has traveled another distance d to point C, will its speed be greater than, less than, or equal to 2vB ? Explain your reasoning.

(I) What is the dot product of A = 2.0x2i 4.0jcj + 5.0k and B = 11.Oi + 2.5xj?

(I) For any vector V = Vx i + Vy\ + Vz k show that Vx = i V, Vy = j V, y7 = k v .

(I) Calculate the angle between the vectors: A = 6.8i - 3.4j - 6.2k and B = 8.2i + 2.3j - 7.0k.

(I) Show that A (-B ) = -A B.

(I) Vector Vi points along the z axis and has magnitude V\ = 75. Vector V2 lies in the xz plane, has magnitude V2 = 58, and makes a 48 angle with the x axis (points below x axis). What is the scalar product Vi V2?

(II) Given the vector A = 3.0i + 1.5j, find a vector B that is perpendicular to A.

(II) A constant force F = (2.0i + 4.0j) N acts on an object as it moves along a straight-line path. If the objects displacement is d = (l.Oi + 5.0j) m, calculate the work done by F using these alternate ways of writing the dot product: (a) W = FdcosO; (b) W = Fx dx + Fy dy .

(II) If A = 9.0i - 8.5j, B = 8.01 + 7.1j + 4.2k, and C = 6.8i - 9.2j, determine (a) A (B + C); (b) (A + C) B; (c) (B + A) C.

(II) Prove that A B = A XBX + A y By + A ZBZ, starting from Eq. 7-2 and using the distributive property (p. 167, proved in Problem 33).

(II) Given vectors A = -4.8i + 6.8j and B = 9.6i + 6.7j, determine the vector C that lies in the xy plane perpendicular to B and whose dot product with A is 20.0.

(II) Show that if two nonparallel vectors have the same magnitude, their sum must be perpendicular to their difference.

(II) Let V = 20.0i + 22.0j - 14.0k. What angles does this vector make with the x, y, and z axes?

(II) Use the scalar product to prove the law o f cosines for a triangle: c2 = a2 + b2 - lab cos 0, where a, b, and c are the lengths of the sides of a triangle and 0 is the angle opposite side c.

(II) Vectors A and B are in the xy plane and their scalar product is 20.0 units. If A makes a 27.4 angle with the x axis and has magnitude A = 12.0 units, and B has magnitude B = 24.0 units, what can you say about the direction of B?

(II) A and B are two vectors in the xy plane that make angles a and /3 with the x axis respectively. Evaluate the scalar product of A and B and deduce the following trigonometric identity: cos (a /3) = cos a cos /3 + sin a sin /3.

(II) Suppose A = l.Oi + l.Oj - 2.0k and B = -l.Oi + l.Oj + 2.0k, (a) what is the angle between these two vectors? (b) Explain the significance of the sign in part (a).

(II) Find a vector of unit length in the xy plane that is perpendicular to 3.0i + 4.0j.

(Ill) Show that the scalar product of two vectors is distributive: A (B + C) = A B + A C. [Hint: Use a diagram showing all three vectors in a plane and indicate dot products on the diagram.]

(I) In pedaling a bicycle uphill, a cyclist exerts a downward force of 450 N during each stroke. If the diameter of the circle traced by each pedal is 36 cm, calculate how much work is done in each stroke.

(II) A spring has k = 65N/m. Draw a graph like that in Fig. 7-11 and use it to determine the work needed to stretch the spring from x = 3.0 cm to x = 6.5 cm, where x = 0 refers to the springs unstretched length.

(II) If the hill in Example 7-2 (Fig. 7-4) was not an even slope but rather an irregular curve as in Fig. 7-23, show that the same result would be obtained as in Example 7-2: namely, that the work done by gravity depends only on the height of the hill and not on its shape or the path taken.

(II) The net force exerted on a particle acts in the positive x direction. Its magnitude increases linearly from zero at x = 0, to 380 N at x = 3.0 m. It remains constant at 380 N from x = 3.0 m to x = 7.0 m, and then decreases linearly to zero at x = 12.0 m. Determine the work done to move the particle from x = 0 to x = 12.0 m graphically, by determining the area under the Fx versus x graph.

(II) If it requires 5.0 J of work to stretch a particular spring by 2.0 cm from its equilibrium length, how much more work will be required to stretch it an additional 4.0 cm?

(II) In Fig. 7-9 assume the distance axis is the x axis and that a = 10.0 m and b = 30.0 m. Estimate the work done by this force in moving a 3.50-kg object from a to b

(II) The force on a particle, acting along the x axis, varies as shown in Fig. 7-24. Determine the work done by this force to move the particle along the x axis: (a) from x = 0.0 to x = 10.0 m; (b) from x = 0.0 to x = 15.0 m.

(II) A child is pulling a wagon down the sidewalk. For 9.0 m the wagon stays on the sidewalk and the child pulls with a horizontal force of 22 N. Then one wheel of the wagon goes off on the grass so the child has to pull with a force of 38 N at an angle of 12 to the side for the next 5.0 m. Finally the wagon gets back on the sidewalk so the child makes the rest of the trip, 13.0 m, with a force of 22 N. How much total work did the child do on the wagon?

(II) The resistance of a packing material to a sharp object penetrating it is a force proportional to the fourth power of the penetration depth*; that is, F = k x 4i. Calculate the work done to force a sharp object a distance d into the material.

(II) The force needed to hold a particular spring compressed an amount x from its normal length is given by F = kx + ax3 + bx4 How much work must be done to compress it by an amount X, starting from x = 0?

(II) At the top of a pole vault, an athlete actually can do work pushing on the pole before releasing it. Suppose the pushing force that the pole exerts back on the athlete is given by F(x) = (1.5 X 102N/m)x - (1.9 X 102N/m2)jt2 acting over a distance of 0.20 m. How much work is done on the athlete?

(II) Consider a force F\ = A /^ /x which acts on an object during its journey along the x axis from x = 0.0 to x = 1.0 m, where A = 2.0 N-m1/2. Show that during this journey, even though F1 is infinite at x = 0.0, the work done on the object by this force is finite.

(II) Assume that a force acting on an object is given by F = axi + by\, where the constants a = 3.0 N*m-1 and b = 4.0 N-m-1. Determine the work done on the object by this force as it moves in a straight line from the origin to r = (lO.Oi + 20.0j) m.

(II) An object, moving along the circumference of a circle with radius R, is acted upon by a force of constant magnitude F. The force is directed at all times at a 3o;>____ _ 30 angle with respect to the tangent to the circle as shown in Fig. 7-25. Determine the work done by this force when the object moves along the half circle from A to B.

(Ill) A 2800-kg space vehicle, initially at rest, falls vertically from a height of 3300 km above the Earths surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earths surface.

(Ill) A 3.0-m-long steel chain is stretched out along the top level of a horizontal scaffold at a construction site, in such a way that 2.0 m of the chain remains on the top level and 1.0 m hangs vertically, Fig. 7-26. At this point, the force on the hanging segment is sufficient to pull the entire chain over the edge. Once the chain is moving, the kinetic friction is so small that it can be neglected. How much work is performed on the chain by the force of gravity as the chain falls from the point where 2.0 m remains on the scaffold to the point where the entire chain has left the scaffold? (Assume that the chain has a linear weight density of 18 N/m.)

(I) At room temperature, an oxygen molecule, with mass of 5.31 X 10_26kg, typically has a kinetic energy of about 6.21 X 10-21 J. How fast is it moving?

(I) (a) If the kinetic energy of a particle is tripled, by what factor has its speed increased? (b) If the speed of a particle is halved, by what factor does its kinetic energy change?

(I) How much work is required to stop an electron (m = 9.11 X 10-31 kg) which is moving with a speed of 1.40 X 106 m/s?

(I) How much work must be done to stop a 1300-kg car traveling at 95 km/h?

(II) Spiderman uses his spider webs to save a runaway train, Fig. 7-27. His web stretches a few city blocks before the 104-kg train comes to a stop. Assuming the web acts like a spring, estimate the spring constant.

(II) A baseball (m = 145 g) traveling 32 m/s moves a fielders glove backward 25 cm when the ball is caught. What was the average force exerted by the ball on the glove?

(II) An 85-g arrow is fired from a bow whose string exerts an average force of 105 N on the arrow over a distance of 75 cm. What is the speed of the arrow as it leaves the bow?

(II) A mass m is attached to a spring which is held stretched a distance x by a force F (Fig. 7-28), and then released. The spring compresses, pulling the mass. Assuming there is no friction, determine the speed of the mass m when the spring returns: (a) to its normal length (x = 0); (b) to half its inal extension (x/2).

(II) If the speed of a car is increased by 50%, by what factor will its minimum braking distance be increased, assuming all else is the same? Ignore the drivers reaction time.

(II) A 1200-kg car rolling on a horizontal surface has speed v = 66 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring constant of the spring?

(II) One car has twice the mass of a second car, but only half as much kinetic energy. When both cars increase their speed by 7.0 m/s, they then have the same kinetic energy. What were the original speeds of the two cars?

(II) A 4.5-kg object moving in two dimensions initially has a velocity vi = (lO.Oi + 20.0j) m/s. A net force F then acts on the object for 2.0 s, after which the objects velocity is v2 = (15.0i + 30.0j) m/s. Determine the work done by F on the object.

(II) A 265-kg load is lifted 23.0 m vertically with an acceleration a = 0.150 g by a single cable. Determine (a) the tension in the cable; (b) the net work done on the load; (c) the work done by the cable on the load; (d) the work done by gravity on the load; (e) the final speed of the load assuming it started from rest.

(II) (a) How much work is done by the horizontal force Fp = 150 N on the 18-kg block of Fig. 7-29 when the force pushes the block 5.0 m up along the 32 frictionless incline? (b) How much work is done by the gravitational force on the block during this displacement? (c) How much work is done by the normal force? (d) What is the speed of the block (assume that it is zero initially) after this displacement? [Hint. Work-energy involves net work done.

(II) Repeat Problem 63 assuming a coefficient of friction = 0.10.

(II) At an accident scene on a level road, investigators measure a cars skid mark to be 98 m long. It was a rainy day and the coefficient of friction was estimated to be 0.38. Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. (Why does the cars mass not matter?)

(II) A 46.0-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 225 N. For the first 11.0 m the floor is frictionless, and for the next 10.0 m the coefficient of friction is 0.20. What is the final speed of the crate after being pulled these 21.0 m?

(II) A train is moving along a track with constant speed relative to the ground. A person on the train holds a ball of mass m and throws it toward the front of the train with a speed v2 relative to the train. Calculate the change in kinetic energy of the ball (a) in the Earth frame of reference, and (b) in the train frame of reference, (c) Relative to each frame of reference, how much work was done on the ball? (d) Explain why the results in part (b) are not the same for the two framesafter all, its the same ball.

(Ill) We usually neglect the mass of a spring if it is small compared to the mass attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length and mass Ms uniformly distributed along the length of the spring. A mass m is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7-30). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed vQ, the midpoint of the spring moves with speed Vq/ 2 . Show that the kinetic energy of the mass plus spring when the mass is moving with velocity v is where M = m + |M S is the effective mass of the system. [Hint: Let D be the total length of the stretched spring. Then the velocity of a mass dm of a spring of length dx located at x is v(x) = v0(x/D). Note also that dm = dx(Ms/D).]

(Ill) An elevator cable breaks when a 925-kg elevator is 22.5 m above the top of a huge spring (k = 8.00 X 104N/m) at the bottom of the shaft. Calculate (a) the work done by gravity on the elevator before it hits the spring; (b) the speed of the elevator just before striking the spring; (c) the amount the spring compresses (note that here work is done by both the spring and gravity).

(a) A 3.0-g locust reaches a speed of 3.0 m/s during its jump. What is its kinetic energy at this speed? (b) If the locust transforms energy with 35% efficiency, how much energy is required for the jump?

In a certain library the first shelf is 12.0 cm off the ground, and the remaining 4 shelves are each spaced 33.0 cm above the previous one. If the average book has a mass of 1.40 kg with a height of 22.0 cm, and an average shelf holds 28 books (standing vertically), how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start?

A 75-kg meteorite buries itself 5.0 m into soft mud. The force between the meteorite and the mud is given by F{x) = (640N/m3)x3, where x is the depth in the mud. What was the speed of the meteorite when it initially impacted the mud?

A 6.10-kg block is pushed 9.25 m up a smooth 37.0 inclined plane by a horizontal force of 75.0 N. If the initial speed of the block is 3.25 m/s up the plane, calculate (a) the initial kinetic energy of the block; (b) the work done by the 75.0-N force; (c) the work done by gravity; (d) the work done by the normal force; (e) the final kinetic energy of the block.

The arrangement of atoms in zinc is an example of hexagonal close-packed structure. Three of the nearest neighbors are found at the following (x, y, z ) coordinates, given in nanometers (l0-9 m): atom 1 is at (0, 0, 0); atom 2 is at (0.230, 0.133, 0); atom 3 is at (0.077, 0.133, 0.247). Find the angle between two vectors: one that connects atom 1 with atom 2 and another that connects atom 1 with atom 3.

Two forces, Fj = (l.50i - 0.80j + 0.70k) N and F2 = (0.70i + 1.20j) N, are applied on a moving object of mass 0.20 kg. The displacement vector produced by the two forces is d = (8.0i + 6.0j + 5.0k) m. What is the work done by the two forces?

The barrels of the 16-in. guns (bore diameter = 16 in. = 41 cm) on the World War II battleship U.S.S. Massachusetts were each 15 m long. The shells each had a mass of 1250 kg and were fired with sufficient explosive force to provide them with a muzzle velocity of 750 m/s. Use the work-energy principle to determine the explosive force (assumed to be a constant) that was applied to the shell within the barrel of the gun. Express your answer in both newtons and in pounds.

A varying force is given by F = Ae~kx, where x is the position; A and k are constants that have units of N and m_1, respectively. What is the work done when x goes from 0.10 m to infinity?

The force required to compress an imperfect horizontal spring an amount x is given by F = 150x + 12x3, where x is in meters and F in newtons. If the spring is compressed 2.0 m, what speed will it give to a 3.0-kg ball held against it and then released?

A force F = (lO.Oi + 9.0j + 12.0k) kN acts on a small object of mass 95 g. If the displacement of the object is d = (5.0i + 4.0j) m, find the work done by the force. What is the angle between F and d?

In the game of paintball, players use guns powered by pressurized gas to propel 33-g gel capsules filled with paint at the opposing team. Game rules dictate that a paintball cannot leave the barrel of a gun with a speed greater than 85m/s. Model the shot by assuming the pressurized gas applies a constant force F to a 33-g capsule over the length of the 32-cm barrel. Determine F (a) using the work-energy principle, and (b) using the kinematic equations (Eqs. 2-12) and Newtons second law.

A softball having a mass of 0.25 kg is pitched horizontally at 110 km/h. By the time it reaches the plate, it may have slowed by 10%. Neglecting gravity, estimate the average force of air resistance during a pitch, if the distance between the plate and the pitcher is about 15 m.

An airplane pilot fell 370 m after jumping from an aircraft without his parachute opening. He landed in a snowbank, creating a crater 1.1 m deep, but survived with only minor injuries. Assuming the pilots mass was 88 kg and his terminal velocity was 45 m/s, estimate: (a) the work done by the snow in bringing him to rest; (b) the average force exerted on him by the snow to stop him; and (c) the work done on him by air resistance as he fell. Model him as a particle.

Many cars have 5 mi/h (8 km/h) bumpers that are designed to compress and rebound elastically without any physical damage at speeds below 8 km/h. If the material of the bumpers permanently deforms after a compression of 1.5 cm, but remains like an elastic spring up to that point, what must be the effective spring constant of the bumper material, assuming the car has a mass of 1050 kg and is tested by ramming into a solid wall?

What should be the spring constant A: of a spring designed to bring a 1300-kg car to rest from a speed of 90 km/h so that the occupants undergo a maximum acceleration of 5.0 g?

Assume a cyclist of weight mg can exert a force on the pedals equal to 0.90 mg on the average. If the pedals rotate in a circle of radius 18 cm, the wheels have a radius of 34 cm, and the front and back sprockets on which the chain runs have 42 and 19 teeth respectively (Fig. 7-31), determine the maximum steepness of hill the cyclist can climb at constant speed. Assume the mass of the bike is 12 kg and that of the rider is 65 kg. Ignore friction. Assume the cyclists average force is always: (a) downward; (b) tangential to pedal motion.

A simple pendulum consists of a small object of mass m (the bob) suspended by a cord of length (Fig. 7-32) of negligible mass. A force F is applied in the horizontal direction (so F = Fi), moving the bob very slowly so the acceleration is essentially zero. (Note that the magnitude of F will need to vary with the angle 6 that the cord makes with the vertical at any moment.) (a) Determine the work done by this force, F, to move the pendulum from 0 = 0 to 6 = 60. (b) Determine the work done by the gravitational force on the bob, Fg = mg, and the work done by the force Ft that the cord exerts on the bob.

A car passenger buckles himself in with a seat belt and holds his 18-kg toddler on his lap. Use the work-energy principle to answer the following questions, (a) While traveling 25 m/s, the driver has to make an emergency stop over a distance of 45 m. Assuming constant deceleration, how much force will the arms of the parent need to exert on the child during this deceleration period? Is this force achievable by an average parent? (b) Now assume that the car (v = 25 m/s) is in an accident and is brought to stop over a distance of 12 m. Assuming constant deceleration, how much force will the parent need to exert on the child? Is this force achievable by an average parent?

As an object moves along the x axis from x = 0.0 m to x = 20.0 m it is acted upon by a force given by F = (100 - (x - 10)2)N. Determine the work done by the force on the object: (a) by first sketching the F vs. x graph and estimating the area under this curve; (b) by evaluating the integral dx.

A cyclist starts from rest and coasts down a 4.0 hill. The mass of the cyclist plus bicycle is 85 kg. After the cyclist has traveled 250 m, (a) what was the net work done by gravity on the cyclist? (b) How fast is the cyclist going? Ignore air resistance.

Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height above the anchor point, he slips and falls under the influence of gravity for a distance 2, after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7-33). Assume a stretchy rope behaves as a spring with spring constant k. (a) Applying the work-energy principle, show that(b) Assuming m = 85 kg, = 8.0 m and k = 850 N/m, determine x/ (the fractional stretch of the rope) and kx /mg (the force that the rope exerts on the climber compared to his own weight) at the moment the climbers fall has been stopped.

A small mass m hangs at rest from a vertical rope of length that is fixed to the ceiling. A force F then pushes on the mass, perpendicular to the taut rope at all times, until the rope is oriented at an angle 6 = 0O and the mass has been raised by a vertical distance h (Fig. 7-34). Assume the forces magnitude F is adjusted so that the mass moves at constant speed along its curved trajectory. Show that the work done by F during this process equals mgh, which is equivalent to the amount of work it takes to slowly lift a mass m straight up by a height h. [Hint: When the angle is increased by dd (in radians), the mass moves along an arc length ds = d6.\

(II) The net force along the linear path of a particle of mass 480 g has been measured at 10.0-cm intervals, starting at jc = 0.0, to be 26.0, 28.5, 28.8, 29.6, 32.8, 40.1, 46.6, 42.2, 48.8,52.6,55.8,60.2,60.6,58.2,53.7,50.3,45.6,45.2,43.2,38.9, 35.1, 30.8, 27.2, 21.0, 22.2, and 18.6, all in newtons. Determine the total work done on the particle over this entire range.

(II) When different masses are suspended from a spring, the spring stretches by different amounts as shown in the Table below. Masses are +1.0 gram. (a) Graph the applied force (in Newtons) versus the stretch (in meters) of the spring, and determine the best-fit straight line. (b) Determine the spring constant (N/m) of the spring from the slope of the best-fit line, (c) If the spring is stretched by 20.0 cm, estimate the force acting on the spring using the best-fit line.