Concept explainers
A particle moves in the xy plane (Fig. P9.30) from the origin to a point having coordinates x = 7.00 m and y = 4.00 m under the influence of a force given by
- a. What is the work done on the particle by the force F if it moves along path 1 (shown in red)?
- b. What is the work done on the particle by the force F if it moves along path 2 (shown in blue)?
- c. What is the work done on the particle by the force F if it moves along path 3 (shown in green)?
- d. Is the force F conservative or nonconservative? Explain.
FIGURE P9.30
In each case, the work is found using the integral of
(a) The work done along path 1, we first need to integrate along
Performing the dot products, we get
Along the first part of this path, y = 0 therefore the first integral equals zero. For the second integral, x is constant and can be pulled out of the integral, and we can evaluate dy.
(b) The work done along path 2 is along
Performing the dot product, we get:
Along the first part of this path, x = 0. Therefore, the first integral equals zero. For the second integral, y is constant and can be pulled out of the integral, and we can evaluate dx.
(c) To find the work along the third path, we first write the expression for the work integral.
At first glance, this appears quite simple, but we can’t integrate
Now, use equation (2) in (1) to express each integral in terms of only one variable.
We can determine the tangent of the angle, which is constant (the angle is the angle of the line with respect to the horizontal).
Insert the value of the tangent and solve the integrals.
(d) Since the work done is not “path-independent”, this is
Figure P9.30ANS
(a)
The work done on the particle by the force
Answer to Problem 30PQ
The work done on the particle by the force
Explanation of Solution
The path 1 followed by the particle is given in Figure P9.30. Given that the coordinate of the final position of the particle is
Write the expression for the work done by a force.
Here,
The path 1 of the particle consist of two parts. Motion from
Use the force vector along with the limits of integration and perform the integration (represent the work done along path 1 as
Perform the dot product to reduce the integral (II).
Along the first part of the path 1,
Conclusion:
Therefore, the work done on the particle by the force
(b)
The work done on the particle by the force
Answer to Problem 30PQ
The work done on the particle by the force
Explanation of Solution
The path 2 followed by the particle is given in Figure P9.30. Given that the coordinate of the final position of the particle is
Equation (I) gives the expression for the work done by a force.
The path 2 of the particle consist of two parts. Motion from
Use the force vector along with the limits of integration and perform the integration (represent the work done along path 2 as
Perform the dot product to reduce the integral (IV).
Along the first part of the path 2,
Conclusion:
Therefore, the work done on the particle by the force
(c)
The work done on the particle by the force
Answer to Problem 30PQ
The work done on the particle by the force
Explanation of Solution
The path 3 followed by the particle is given in Figure P9.30. Given that the coordinate of the final position of the particle is
Equation (I) gives the expression for the work done by a force.
Write equation (I) in terms of
Use the
The path 3 of the particle starts from
Here, both
Write the expression relating
Solve equation (IX) for
Solve equation (IX) for
Use equation (X) and (XI) in (VIII).
Compute
Use equation (XIII) in (XII) and perform the integral.
Conclusion:
Therefore, the work done on the particle by the force
(d)
Whether the force
Answer to Problem 30PQ
The force
Explanation of Solution
From part (a), (b) and (c) it is found that the work done by the force
Conclusion:
Therefore, the force
Want to see more full solutions like this?
Chapter 9 Solutions
Student Solutions Manual For Katz's Physics For Scientists And Engineers: Foundations And Connections, Volume 1
- A nonconstant force is exerted on a particle as it moves in the positive direction along the x axis. Figure P9.26 shows a graph of this force Fx versus the particles position x. Find the work done by this force on the particle as the particle moves as follows. a. From xi = 0 to xf = 10.0 m b. From xi = 10.0 to xf = 20.0 m c. From xi = 0 to xf = 20.0 m FIGURE P9.26 Problems 26 and 27.arrow_forwardA particle moves in one dimension under the action of a conservative force. The potential energy of the system is given by the graph in Figure P8.55. Suppose the particle is given a total energy E, which is shown as a horizontal line on the graph. a. Sketch bar charts of the kinetic and potential energies at points x = 0, x = x1, and x = x2. b. At which location is the particle moving the fastest? c. What can be said about the speed of the particle at x = x3? FIGURE P8.55arrow_forwardA force acting on a particle moving in the xy plane is given by F=(2yi+x2j), where F is in newtons and x and y are in meters. The particle moves from the origin to a final position having coordinates x = 5.00 m and y = 5.00 m as shown in Figure P7.31. Calculate the work done by F on the particle as it moves along (a) the purple path, (b) the red path, and (c) the blue path, (d) Is F conservative or nonconservative? (e) Explain your answer to part (d). Figure P7.31arrow_forward
- A particle is subject to a force Fx that varies with position as shown in Figure P7.9. Find the work done by the force on the particle as it moves (a) from x = 0 to x = 5.00 m, (b) from x = 5.00 m to x = 10.0 m, and (c) from x = 10.0 m to x = 15.0 m. (d) What is the total work done by the force over the distance x = 0 to x = 15.0 m?arrow_forward(a) A force F=(4xi+3yj), where F 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 = 5.00 m. Find the work W=Fdr done by the force on the object. (b) What If? Find the work W=Fdr done by the force on the object if it moves from the origin to (5.00 m, 5.00 m) along a straightline path making an angle of 45.0 with the positive x axis. Is the work done by this force dependent on the path taken between the initial and final points?arrow_forwardA small object is attached to two springs of the same length l, but with different spring constants k1 and k2 as shown in Figure P9.31. Initially, both springs are relaxed. The object is then displaced straight along the x axis from xi to xf. Find an expression for the work done by the springs on the object.arrow_forward
- Figure P9.65A shows a crate attached to a rope that is extended over an ideal pulley. Boris pulls on the other end of the rope with a constant force until the crate has risen a total distance of 6.53 m (Fig. P9.65B). If the crate has a mass of 81.36 kg, what is the average power exerted by Boris, assuming he accomplishes the task in 5.33 s? FIGURE P9.65arrow_forwardSuppose the ski patrol lowers a rescue sled and victim, having a total mass of 90.0 kg, down a 60.0° slope at constant speed, as shown in Figure 7.37. The coefficient of friction between the sled and the snow is 0.100. (a) How much work is done by friction as the sled moves 30.0 m along the hill? (b) How much work is done by the rope on the sled in this distance? (c) What is the work done by the gravitational force on the sled? (d) What is the total work done?arrow_forwardA helicopter rescues a trapped person of mass m = 65.0 kg from a flooded river by lifting the person vertically upward using a winch and rope. The person is pulled 12.0 m into the helicopter with a constant force that is 15% greater than the persons weight. a. Find the work done by each of the forces acting on the person. b. Assuming the survivor starts from rest, determine his speed upon reaching the helicopter.arrow_forward
- A constant force of magnitude 4.75 N is exerted on an object. The forces direction is 60.0 counterclockwise from the positive x axis in the xy plane, and the objects displacement is r=(4.22.1j+1.6k)m. Calculate the work done by this force.arrow_forwardA cat’s crinkle ball toy of mass 15 g is thrown straight up with an initial speed of 3 m/s. Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the ball as it leaves the hand? (b) How much work is done by the gravitational force during the ball’s rise to its peak? (c) What is the change in the gravitational potential energy of the ball during the rise to its peak? (d) If the gravitational potential energy is taken to be zero at the point where it leaves your hand, what is the gravitational potential energy when it reaches the maximum height? (e) What if the gravitational potential energy is taken to be zero at the maximum height the ball reaches, what would the gravitational potential energy be when it leaves the hand? (f) What is the maximum height the ball reaches?arrow_forwardA block of mass m = 2.50 kg is pushed a distance d = 2.20 m along a frictionless, horizontal table by a constant applied force of magnitude F = 16.0 N directed at an angle = 25.0 below the horizontal as shown in Figure P6.3. Determine the work done on the block by (a) the applied force, (b) the normal force exerted by the table, (c) the gravitational force, and (d) the net force on the block. Figure P6.3arrow_forward
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPrinciples of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
- Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningCollege PhysicsPhysicsISBN:9781938168000Author:Paul Peter Urone, Roger HinrichsPublisher:OpenStax CollegePhysics for Scientists and Engineers, Technology ...PhysicsISBN:9781305116399Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning