The spring constant of an automotive suspension spring

Alex and John are loading identical cabinets onto a truck. Alex lifts his cabinet straight up from the ground to the bed of the truck, whereas John slides his cabinet up a rough ramp to the truck. Which statement is correct about the work done on the cabinet Earth system? (a) Alex and John do the same amount of work. (b) Alex does more work than John. (c) John does more work than Alex. (d) None of those statements is necessarily true because the force of friction is unknown. (e) None of those statements is necessarily true because the angle of the incline is unknown.

If the net work done by external forces on a particle is zero, which of the following statements about the particle must be true? (a) Its velocity is zero. (b) Its velocity is decreased. (c) Its velocity is unchanged. (d) Its speed is unchanged. (e) More information is needed.

A worker pushes a wheelbarrow with a horizontal force of 50 N on level ground over a distance of 5.0 m. If a friction force of 43 N acts on the wheelbarrow in a direction opposite that of the worker, what work is done on the wheelbarrow by the worker? (a) 250 J (b) 215 J (c) 35 J (d) 10 J (e) None of those answers is correct.

A cart is set rolling across a level table, at the same speed on every trial. If it runs into a patch of sand, the cart exerts on the sand an average horizontal force of 6 N and travels a distance of 6 cm through the sand as it comes to a stop. If instead the cart runs into a patch of gravel on which the cart exerts an average horizontal force of 9 N, how far into the gravel will the cart roll before stopping? (a) 9 cm (b) 6 cm (c) 4 cm (d) 3 cm (e) none of those answers

Let N^ represent the direction horizontally north, NE represent northeast (halfway between north and east), and so on. Each direction specification can be thought of as a unit vector. Rank from the largest to the smallest the following dot products. Note that zero is larger than a negative number. If two quantities are equal, display that fact in your ranking. (a) N^ ?N^ (b) N^ ?NE (c) N^ ? S^ (d) N^ ?E^ (e) SE? S

Is the work required to be done by an external force on an object on a frictionless, horizontal surface to accelerate it from a speed v to a speed 2v (a) equal to the work required to accelerate the object from v 5 0 to v, (b) twice the work required to accelerate the object from v 5 0 to v, (c) three times the work required to accelerate the object from v 5 0 to v, (d) four times the work required to accelerate the object from 0 to v, or (e) not known without knowledge of the acceleration?

A block of mass m is dropped from the fourth floor of an office building and hits the sidewalk below at speed v. From what floor should the block be dropped to double that impact speed? (a) the sixth floor (b) the eighth floor (c) the tenth floor (d) the twelfth floor (e) the sixteenth floor

As a simple pendulum swings back and forth, the forces acting on the suspended object are (a) the gravitational force, (b) the tension in the supporting cord, and (c) air resistance. (i) Which of these forces, if any, does no work on the pendulum at any time? (ii) Which of these forces does negative work on the pendulum at all times during its motion?

Bullet 2 has twice the mass of bullet 1. Both are fired so that they have the same speed. If the kinetic energy of bullet 1 is K, is the kinetic energy of bullet 2 (a) 0.25K, (b) 0.5K, (c) 0.71K, (d) K, or (e) 2K?

Figure OQ7.10 shows a light extended spring exerting a force Fs to the left on a block. (i) Does the block exert a force on the spring? Choose every correct answer. (a) No, it doesnt. (b) Yes, it does, to the left. (c) Yes, it does, to the right. (d) Yes, it does, and its magnitude is larger than Fs. (e) Yes, it does, and its magnitude is equal to Fs. (ii) Does the spring exert a force on the wall? Choose your answers from the same list (a) through (e).

If the speed of a particle is doubled, what happens to its kinetic energy? (a) It becomes four times larger. (b) It becomes two times larger. (c) It becomes !2 times larger. (d) It is unchanged. (e) It becomes half as large.

Mark and David are loading identical cement blocks onto Davids pickup truck. Mark lifts his block straight up from the ground to the truck, whereas David slides his block up a ramp containing frictionless rollers. Which statement is true about the work done on the blockEarth system? (a) Mark does more work than David. (b) Mark and David do the same amount of work. (c) David does more work than Mark. (d) None of those statements is necessarily true because the angle of the incline is unknown. (e) None of those statements is necessarily true because the mass of one block is not given.

(i) Rank the gravitational accelerations you would measure for the following falling objects: (a) a 2-kg object 5 cm above the floor, (b) a 2-kg object 120 cm above the floor, (c) a 3-kg object 120 cm above the floor, and (d) a 3-kg object 80 cm above the floor. List the one with the largest magnitude of acceleration first. If any are equal, show their equality in your list. (ii) Rank the gravitational forces on the same four objects, listing the one with the largest magnitude first. (iii) Rank the gravitational potential energies (of the objectEarth system) for the same four objects, largest first, taking y 5 0 at the floor.

A certain spring that obeys Hookes law is stretched by an external agent. The work done in stretching the spring by 10 cm is 4 J. How much additional work is required to stretch the spring an additional 10 cm? (a) 2 J (b) 4 J (c) 8 J (d) 12 J (e) 16 J

A cart is set rolling across a level table, at the same speed on every trial. If it runs into a patch of sand, the cart exerts on the sand an average horizontal force of 6 N and travels a distance of 6 cm through the sand as it comes to a stop. If instead the cart runs into a patch of flour, it rolls an average of 18 cm before stopping. What is the average magnitude of the horizontal force the cart exerts on the flour? (a) 2 N (b) 3 N (c) 6 N (d) 18 N (e) none of those answers

An ice cube has been given a push and slides without friction on a level table. Which is correct? (a) It is in stable equilibrium. (b) It is in unstable equilibrium. (c) It is in neutral equilibrium. (d) It is not in equilibrium.

When a 4.00-kg object is hung vertically on a certain light spring that obeys Hookes law, the spring stretches 2.50 cm. If the 4.00-kg object is removed, (a) how far will the spring stretch if a 1.50-kg block is hung on it? (b) How much work must an external agent do to stretch the same spring 4.00 cm from its unstretched position?

Hookes law describes a certain light spring of unstretched length 35.0 cm. When one end is attached to the top of a doorframe and a 7.50-kg object is hung from the other end, the length of the spring is 41.5 cm. (a) Find its spring constant. (b) The load and the spring are taken down. Two people pull in opposite directions on the ends of the spring, each with a force of 190 N. Find the length of the spring in this situation

An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from zero to 230 N. (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do on the string in drawing the bow?

A light spring with spring constant 1 200 N/m is hung from an elevated support. From its lower end hangs a second light spring, which has spring constant 1 800 N/m. An object of mass 1.50 kg is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.

A light spring with spring constant k1 is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant k2. An object of mass m is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system.

Express the units of the force constant of a spring in SI fundamental units.

Express the units of the force constant of a spring in SI fundamental units.

A light spring with force constant 3.85 N/m is compressed by 8.00 cm as it is held between a 0.250-kg block on the left and a 0.500-kg block on the right, both resting on a horizontal surface. The spring exerts a force on each block, tending to push the blocks apart. The blocks are simultaneously released from rest. Find the acceleration with which each block starts to move, given that the coefficient of kinetic friction between each block and the surface is (a) 0, (b) 0.100, and (c) 0.462.

A small particle of mass m is pulled to the top of a frictionless halfcylinder (of radius R) by a light cord that passes over the top of the cylinder as illustrated in Figure P7.25. (a) Assuming the particle moves at a constant speed, show that F 5 mg cos u. Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times. (b) By directly integrating W 5 e F S ? d r S, find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder

The force acting on a particle is Fx 5 (8x 2 16), where F is in newtons and x is in meters. (a) Make a plot of this force versus x from x 5 0 to x 5 3.00 m. (b) From your graph, find the net work done by this force on the particle as it moves from x 5 0 to x 5 3.00 m.

When different loads hang on a spring, the spring stretches to different lengths as shown in the following table. (a) Make a graph of the applied force versus the extension of the spring. (b) By least-squares fitting, determine the straight line that best fits the data. (c) To complete part (b), do you want to use all the data points, or should you ignore some of them? Explain. (d) From the slope of the best-fit line, find the spring constant k. (e) If the spring is extended to 105 mm, what force does it exert on the suspended object?

A 100-g bullet is fired from a rifle having a barrel 0.600 m long. Choose the origin to be at the location where the bullet begins to move. Then the force (in newtons) exerted by the expanding gas on the bullet is 15 000 1 10 000x 2 25 000x2, where x is in meters. (a) Determine the work done by the gas on the bullet as the bullet travels the length of the barrel. (b) What If? If the barrel is 1.00 m long, how much work is done, and (c) how does this value compare with the work calculated in part (a)?

A force F S 5 14x i ^ 1 3yj ^2, where F S is in newtons and x and y are in meters, acts on an object as the object moves in the x direction from the origin to x 5 5.00 m. Find the work W 5 e F S ? d r S done by the force on the object.

Review. The graph in Figure P7.30 specifies a functional relationship between the two variables u and v. (a) Find eb a u dv. (b) Find ea b u dv. (c) Find eb a v du.

A 3.00-kg object has a velocity 16.00 i ^ 2 2.00 j ^2 m/s. (a) What is its kinetic energy at this moment? (b) What is the net work done on the object if its velocity changes to 18.00 i ^ 1 4.00 j ^2 m/s? (Note: From the definition of the dot product, v 2 5 v S ? v S .)

A worker pushing a 35.0-kg wooden crate at a constant speed for 12.0 m along a wood floor does 350 J of work by applying a constant horizontal force of magnitude F on the crate. (a) Determine the value of F. (b) If the worker now applies a force greater than F, describe the subsequent motion of the crate. (c) Describe what would happen to the crate if the applied force is less than F

A 0.600-kg particle has a speed of 2.00 m/s at point A and kinetic energy of 7.50 J at point B. What is (a) its kinetic energy at A, (b) its speed at B, and (c) the net work done on the particle by external forces as it moves from A to B?

A 4.00-kg particle is subject to a net force that varies with position as shown in Figure P7.15. The particle starts from rest at x 5 0. What is its speed at (a) x 5 5.00 m, (b) x 5 10.0 m, and (c) x 5 15.0 m?

A 2 100-kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls 5.00 m before coming into contact with the top of the beam, and it drives the beam 12.0 cm farther into the ground before coming to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest.

Review. In an electron microscope, there is an electron gun that contains two charged metallic plates 2.80 cm apart. An electric force accelerates each electron in the beam from rest to 9.60% of the speed of light over this distance. (a) Determine the kinetic energy of the electron as it leaves the electron gun. Electrons carry this energy to a phosphorescent viewing screen where the microscopes image is formed, making it glow. For an electron passing between the plates in the electron gun, determine (b) the magnitude of the constant electric force acting on the electron, (c) the acceleration of the electron, and (d) the time interval the electron spends between the plates.

Review. You can think of the workkinetic energy theorem as a second theory of motion, parallel to Newtons laws in describing how outside influences affect the motion of an object. In this problem, solve parts (a), (b), and (c) separately from parts (d) and (e) so you can compare the predictions of the two theories. A 15.0-g bullet is accelerated from rest to a speed of 780 m/s in a rifle barrel of length 72.0 cm. (a) Find the kinetic energy of the bullet as it leaves the barrel. (b) Use the workkinetic energy theorem to find the net work that is done on the bullet. (c) Use your result to part (b) to find the magnitude of the average net force that acted on the bullet while it was in the barrel. (d) Now model the bullet as a particle under constant acceleration. Find the constant acceleration of a bullet that starts from rest and gains a speed of 780 m/s over a distance of 72.0 cm. (e) Modeling the bullet as a particle under a net force, find the net force that acted on it during its acceleration. (f) What conclusion can you draw from comparing your results of parts (c) and (e)?

Review. A 7.80-g bullet moving at 575 m/s strikes the hand of a superhero, causing the hand to move 5.50 cm in the direction of the bullets velocity before stopping. (a) Use work and energy considerations to find the average force that stops the bullet. (b) Assuming the force is constant, determine how much time elapses between the moment the bullet strikes the hand and the moment it stops moving

Review. A 5.75-kg object passes through the origin at time t 5 0 such that its x component of velocity is 5.00 m/s and its y component of velocity is 23.00 m/s. (a) What is the kinetic energy of the object at this time? (b) At a later time t 5 2.00 s, the particle is located at x 5 8.50 m and y 5 5.00 m. What constant force acted on the object during this time interval? (c) What is the speed of the particle at t 5 2.00 s?

A 1 000-kg roller coaster car is initially at the top of a rise, at point A. It then moves 135 ft, at an angle of 40.08 below the horizontal, to a lower point B. (a) Choose the car at point B to be the zero configuration for gravitational potential energy of the roller coaster Earth system. Find the potential energy of the system when the car is at points A and B, and the change in potential energy as the car moves between these points. (b) Repeat part (a), setting the zero configuration with the car at point A.

A 0.20-kg stone is held 1.3 m above the top edge of a water well and then dropped into it. The well has a depth of 5.0 m. Relative to the configuration with the stone at the top edge of the well, what is the gravitational potential energy of the stoneEarth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well?

A 400-N child is in a swing that is attached to a pair of ropes 2.00 m long. Find the gravitational potential energy of the childEarth system relative to the childs lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.08 angle with the vertical, and (c) the child is at the bottom of the circular arc.

A 4.00-kg particle moves from the origin to position C, having coordinates x 5 5.00 m and y 5 5.00 m (Fig. P7.43). One force on the particle is the gravitational force acting in the negative y direction. Using Equation 7.3, calculate the work done by the gravitational force on the particle as it goes from O to C along (a) the purple path, (b) the red path, and (c) the blue path. (d) Your results should all be identical. Why?

(a) Suppose a constant force acts on an object. The force does not vary with time or with the position or the velocity of the object. Start with the general definition for work done by a force W 5 3 f i F S ? d r S and show that the force is conservative. (b) As a special case, suppose the force F S 5 13 i ^ 1 4j ^2 N acts on a particle that moves from O to C in Figure P7.43. Calculate the work done by F S on the particle as it moves along each one of the three paths shown in the figure and show that the work done along the three paths is identical.

A force acting on a particle moving in the xy plane is given by F S 5 12yi ^ 1 x 2 j ^2, where F S is in newtons and x and y are in meters. The particle moves from the origin to a final position having coordinates x 5 5.00 m and y 5 5.00 m as shown in Figure P7.43. Calculate the work done by F S on the particle as it moves along (a) the purple path, (b) the red path, and (c) the blue path. (d) Is F S conservative or nonconservative? (e) Explain your answer to part (d).

An object moves in the xy plane in Figure P7.43 and experiences a friction force with constant magnitude 3.00 N, always acting in the direction opposite the objects velocity. Calculate the work that you must do to slide the object at constant speed against the friction force as the object moves along (a) the purple path O to A followed by a return purple path to O, (b) the purple path O to C followed by a return blue path to O, and (c) the blue path O to C followed by a return blue path to O. (d) Each of your three answers should be nonzero. What is the significance of this observation?

The potential energy of a system of two particles separated by a distance r is given by U(r) 5 A/r, where A is a constant. Find the radial force F S r that each particle exerts on the other

Why is the following situation impossible? A librarian lifts a book from the ground to a high shelf, doing 20.0 J of work in the lifting process. As he turns his back, the book falls off the shelf back to the ground. The gravitational force from the Earth on the book does 20.0 J of work on the book while it falls. Because the work done was 20.0 J 1 20.0 J 5 40.0 J, the book hits the ground with 40.0 J of kinetic energy.

A potential energy function for a system in which a two-dimensional force acts is of the form U 5 3x3 y 2 7x. Find the force that acts at the point (x, y)

A single conservative force acting on a particle within a system varies as F S 5 12Ax 1 Bx 2 2i ^, where A and B are constants, F S is in newtons, and x is in meters. (a) Calculate the potential energy function U(x) associated with this force for the system, taking U 5 0 at x 5 0. Find (b) the change in potential energy and (c) the change in kinetic energy of the system as the particle moves from x 5 2.00 m to x 5 3.00 m.

A single conservative force acts on a 5.00-kg particle within a system due to its interaction with the rest of the system. The equation Fx 5 2x 1 4 describes the force, where Fx is in newtons and x is in meters. As the particle moves along the x axis from x 5 1.00 m to x 5 5.00 m, calculate (a) the work done by this force on the particle, (b) the change in the potential energy of the system, and (c) the kinetic energy the particle has at x 5 5.00 m if its speed is 3.00 m/s at x 5 1.00 m.

For the potential energy curve shown in Figure P7.52, (a) determine whether the force Fx is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for Fx versus x from x 5 0 to x 5 9.5 m.

A right circular cone can theoretically be balanced on a horizontal surface in three different ways. Sketch these three equilibrium configurations and identify them as positions of stable, unstable, or neutral equilibrium.

The potential energy function for a system of particles is given by U(x) 5 2x3 1 2x2 1 3x, where x is the position of one particle in the system. (a) Determine the force Fx on the particle as a function of x. (b) For what values of x is the force equal to zero? (c) Plot U(x) versus x and Fx versus x and indicate points of stable and unstable equilibrium.

Review. A baseball outfielder throws a 0.150-kg baseball at a speed of 40.0 m/s and an initial angle of 30.08 to the horizontal. What is the kinetic energy of the baseball at the highest point of its trajectory?

A particle moves along the x axis from x 5 12.8 m to x 5 23.7 m under the influence of a force F 5 375 x 3 1 3.75x where F is in newtons and x is in meters. Using numerical integration, determine the work done by this force on the particle during this displacement. Your result should be accurate to within 2%.

Two identical steel balls, each of diameter 25.4 mm and moving in opposite directions at 5 m/s, run into each other head-on and bounce apart. Prior to the collision, one of the balls is squeezed in a vise while precise measurements are made of the resulting amount of compression. The results show that Hookes law is a fair model of the balls elastic behavior. For one datum, a force of 16 kN exerted by each jaw of the vise results in a 0.2-mm reduction in the diameter. The diameter returns to its original value when the force is removed. (a) Modeling the ball as a spring, find its spring constant. (b) Does the interaction of the balls during the collision last only for an instant or for a nonzero time interval? State your evidence. (c) Compute an estimate for the kinetic energy of each of the balls before they collide. (d) Compute an estimate for the maximum amount of compression each ball undergoes when the balls collide. (e) Compute an order-of-magnitude estimate for the time interval for which the balls are in contact. (In Chapter 15, you will learn to calculate the contact time interval precisely.)

When an object is displaced by an amount x from stable equilibrium, a restoring force acts on it, tending to return the object to its equilibrium position. The magnitude of the restoring force can be a complicated function of x. In such cases, we can generally imagine the force function F(x) to be expressed as a power series in x as F(x) 5 2(k1x 1 k2x2 1 k3x3 1 . . . ). The first term here is just Hookes law, which describes the force exerted by a simple spring for small displacements. For small excursions from equilibrium, we generally ignore the higher-order terms, but in some cases it may be desirable to keep the second term as well. If we model the restoring force as F 5 2(k1x 1 k2x2), how much work is done on an object in displacing it from x 5 0 to x 5 xmax by an applied force 2F ?

A 6 000-kg freight car rolls along rails with negligible friction. The car is brought to rest by a combination of two coiled springs as illustrated in Figure P7.59. Both springs are described by Hookes law and have spring constants k1 5 1 600 N/m and k2 5 3 400 N/m. After the first spring compresses a distance of 30.0 cm, the second spring acts with the first to increase the force as additional compression occurs as shown in the graph. The car comes to rest 50.0 cm after first contacting the two-spring system. Find the cars initial speed.

Why is the following situation impossible? In a new casino, a supersized pinball machine is introduced. Casino advertising boasts that a professional basketball player can lie on top of the machine and his head and feet will not hang off the edge! The ball launcher in the machine sends metal balls up one side of the machine and then into play. The spring in the launcher (Fig. P7.60) has a force constant of 1.20 N/cm. The surface on which the ball moves is inclined u 5 10.08 with respect to the horizontal. The spring is initially compressed its maximum distance d 5 5.00 cm. A ball of mass 100 g is projected into play by releasing the plunger. Casino visitors find the play of the giant machine quite exciting

Review. Two constant forces act on an object of mass m 5 5.00 kg moving in the xy plane as shown in Figure P7.61. Force F S 1 is 25.0 N at 35.08, and force F S 2 is 42.0 N at 1508. At time t 5 0, the object is at the origin and has velocity 14.00 i ^ 1 2.50 j ^2 m/s. (a) Express the two forces in unit-vector notation. Use unit-vector notation for your other answers. (b) Find the total force exerted on the object. (c) Find the objects acceleration. Now, considering the instant t 5 3.00 s, find (d) the objects velocity, (e) its position, (f) its kinetic energy from 1 2mvf 2 , and (g) its kinetic energy from 1 2mvi 2 1 g F S ?Dr S. (h) What conclusion can you draw by comparing the answers to parts (f) and (g)?

The spring constant of an automotive suspension spring increases with increasing load due to a spring coil that is widest at the bottom, smoothly tapering to a smaller diameter near the top. The result is a softer ride on normal road surfaces from the wider coils, but the car does not bottom out on bumps because when the lower coils collapse, the stiffer coils near the top absorb the load. For such springs, the force exerted by the spring can be empirically found to be given by F 5 axb . For a tapered spiral spring that compresses 12.9 cm with a 1 000-N load and 31.5 cm with a 5 000-N load, (a) evaluate the constants a and b in the empirical equation for F and (b) find the work needed to compress the spring 25.0 cm.

An inclined plane of angle u 5 20.08 has a spring of force constant k 5 500 N/m fastened securely at the bottom so that the spring is parallel to the surface as shown in Figure P7.63. A block of mass m 5 2.50 kg is placed on the plane at a distance d 5 0.300 m from the spring. From this position, the block is projected downward toward the spring with speed v 5 0.750 m/s. By what distance is the spring compressed when the block momentarily comes to rest?

An inclined plane of angle u has a spring of force constant k fastened securely at the bottom so that the spring is parallel to the surface. A block of mass m is placed on the plane at a distance d from the spring. From this position, the block is projected downward toward the spring with speed v as shown in Figure P7.63. By what distance is the spring compressed when the block momentarily comes to rest?

(a) Take U 5 5 for a system with a particle at position x 5 0 and calculate the potential energy of the system as a function of the particle position x. The force on the particle is given by (8e22x ) i ^. (b) Explain whether the force is conservative or nonconservative and how you can tell.

A particle of mass m 5 1.18 kg is attached between two identical springs on a frictionless, horizontal tabletop. Both springs have spring constant k and are initially unstressed, and the particle is at x 5 0. (a) The particle is pulled a distance x along a direction perpendicular to the initial configuration of the springs as shown in Figure P7.66. Show that the force exerted by the springs on the particle is F S 5 22kx a1 2 L "x 2 1 L2 b i (b) Show that the potential energy of the system is U 1x2 5 kx 2 1 2kL1L 2 "x 2 1 L2 2 (c) Make a plot of U(x) versus x and identify all equilibrium points. Assume L 5 1.20 m and k 5 40.0 N/m. (d) If the particle is pulled 0.500 m to the right and then released, what is its speed when it reaches x 5 0?

Review. A light spring has unstressed length 15.5 cm. It is described by Hookes law with spring constant 4.30 N/m. One end of the horizontal spring is held on a fixed vertical axle, and the other end is attached to a puck of mass m that can move without friction over a horizontal surface. The puck is set into motion in a circle with a period of 1.30 s. (a) Find the extension of the spring x as it depends on m. Evaluate x for (b) m 5 0.070 0 kg, (c) m 5 0.140 kg, (d) m 5 0.180 kg, and (e) m 5 0.190 kg. (f) Describe the pattern of variation of x as it depends on m.