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Chapter 7: Q61P (page 175)

How much work is done by a force $\overset{鈫�}{F} = (2 x N) \hat{i} + (3 N) \hat{j}$ ,with x in meters, that moves a particle from a position $\overset{鈫�}{r_{i}} = (2 m) \hat{i} + (3 m) \hat{j}$ to a position $\overset{鈫�}{r_{f}} = - (4 m) \hat{i} + (3 m) \hat{j}$ ?

Short Answer

Expert verified

Work done by the applied force is $- 6 J$

Step by step solution

01

Given information

It is given that,

$F_{a p p} = (2 x N) \hat{i} + (3 N) \hat{j} \overset{鈫�}{r_{i}} = (2 m) \hat{i} + (3 m) \hat{j} \overset{鈫�}{r_{f}} = - (4 m) \hat{i} - (3 m) \hat{j}$

02

Determining the concept

This problem deals with the variable forces. When the direction and the amount of force vary throughout the motion, variable forces occur. To solve this problem, usetheconcept of variable forces. Work done due to variable forces can be found by using integration.

Formula:

The work done is given by,

$W = 鈭� F_{x} . d x + 鈭� F_{y} . d y$

Where,

$F_{x,} F_{y}$ Are the forces in x and y direction, dx and dy are small displacements.

03

Determining the work done by the applied force (Wapp).

Using the above formula, writethetotal work done as,

$W = {鈭�}_{x i}^{x f} F_{x} d x + {鈭�}_{y i}^{y f} F_{y} d y$

Substituting the value of forces and from the given data,

$W = {鈭�}_{2}^{- 4} 2 脳 d x + {鈭�}_{3}^{- 3} 3 d y W = {[x^{2}]}_{2}^{- 4} + {[3 y]}_{3}^{- 3} W = 16 - 4 - 9 - 9 W = - 6 J$

Work done by the applied force is $- 6 J$

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Most popular questions from this chapter

A spring with a spring constant of $^{180 N / cm}$ has a cage attached to its free end. (a) How much work does the spring force do on the cage when the spring is stretched from its relaxed length by $^{7.60 mm}$ ? (b) How much additional work is done by the spring force when the spring is stretched by an additional $^{7.60 mm}$ ?

Figure 7-40 gives the acceleration of a 2.00 kg particle as an applied force $\overset{鈫�}{F_{a}}$ moves it from rest along an x axis from $x = 0 to x = 9.0 m$ . The scale of the figure鈥檚 vertical axis is set by $a_{s} = 6.0 m / s^{2}$ . How much work has the force done on the particle when the particle reaches (a) x=4.0 m, (b) x=7.0 m, and (c) x=9.0 m? What is the particle鈥檚 speed and direction of travel when it reaches (d) x=4.0 m, (e) x=7.0 m, and (f) x=9.0 m ?

In Fig. 7-20 , a greased pig has a choice of three frictionless slides along which to slide to the ground. Rank the slides according to how much work the gravitational force does on the pig during the descent, greatest first

(a) In 1975 the roof of Montreal鈥檚 Velodrome, with a weight of 360 kN, was lifted by 10 cm so that it could be centered. How much work was done on the roof by the forces making the lift? (b) In 1960 a Tampa, Florida, mother reportedly raised one end of a car that had fallen onto her son when a jack failed. If her panic lift effectively raised 4000 N (about of the car鈥檚 weight) by 5.0 cm, how much work did her force do on the car?

A can of bolts and nuts is pushed 2.00 m along an x axis by a broom along the greasy (frictionless) floor of a car repair shop in a version of shuffleboard. Figure 7-26 gives the work Wdone on the can by the constant horizontal force from the broom, versus the can鈥檚 position x. The scale of the figure鈥檚 vertical axis is set by W_S=6 J. (a) What is the magnitude of that force? (b) If the can had an initial kinetic energy of 3.00 J, moving in the positive direction of the x axis, what is its kinetic energy at the end of the 2.00 m?

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