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Work is the ____________ transferred to or from an object by a force acting on the object.

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True or False An interpretation of the line integral \(\int_{C}\mathbf{F\,{\cdot}\,}d\mathbf{r}\) is the work done by the force \(\mathbf{F}\) in moving an object along a piecewise-smooth curve \(C\).

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True or False The work done by a force \(\mathbf{F}\) in moving an object from \(A\) to \(B\) equals the change in the potential energy of the object from \(A\) to \(B\).

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Multiple Choice In a conservative force field \(\mathbf{F}\), the work \(W\) done by \(\mathbf{F}\) in moving an object along a closed, piecewise smooth curve \(C\) is [(a) a maximum, (b) a minimum, (c) \(\left\Vert \mathbf{F}\right\Vert ,\) (d) \(0\)].

1007

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Kinetic energy \(K\) is the energy of ____________; potential energy \(U\) is the energy of ____________.

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True or False In a conservative force field \(\mathbf{F}\), the difference of the potential energy \(U\) and the kinetic energy \(K\) equals \(0\).

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\(\mathbf{F}=y\mathbf{i}+x\mathbf{j}\) along \(\mathbf{r}(t)=t \mathbf{i}+t^{2}\mathbf{j}\) from \(t=0\) to \(t=1\)

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\(\mathbf{F}=xy\mathbf{i}+y^{2}\mathbf{j}\) along \(\mathbf{r} (t)=t\mathbf{i}+t^{2}\mathbf{j}\) from \(t=0\) to \(t=1\)

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\(\mathbf{F}=(x-2y)\mathbf{i}+xy\mathbf{j}\) along \(\mathbf{r} (t)=3\cos t\,\mathbf{i}+2\sin t\,\mathbf{j}\) from \(t=0\) to \(t=\dfrac{\pi }{2}\)

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\(\mathbf{F}=x\mathbf{i}-y\mathbf{j}\) along \(\mathbf{r} (t)=\cos t\,\mathbf{i}+\sin t\,\mathbf{j}\) from \(t=0\) to \(t=2\pi\)

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\(\mathbf{F}=x^{2}y\mathbf{i}+(x^{2}-y^{2})\mathbf{j}\) along \(y=2x^{2}\) from \((0,0)\) to \((1,2)\)

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\(\mathbf{F}=y\sin x\,\mathbf{i}-x\cos y\,\mathbf{j}\) along \(y=x\) from \((0,0)\) to \((1,1)\)

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\(\mathbf{F}=(y-x^{2})\mathbf{i}+x\mathbf{j}\) along the upper half of the circle \(x^{2}+y^{2}=1\) from \((1,0)\) to \((-1,0)\)

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\(\mathbf{F}=(y-x^{2})\mathbf{i}+x\mathbf{j}\) along the line segments joining \((1,0)\) to \((1,1)\) to \((-1,1)\) to \((-1,0)\)

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\(\mathbf{F}=y^{3}\mathbf{i}+x^{3}\mathbf{j}\) along the ellipse \(\mathbf{r}(t)=a\cos t\,\mathbf{i}+b\sin t\,\mathbf{j}\) from \(t=0\) to \(t=2\pi\)

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\(\mathbf{F}=-y^{2}\mathbf{i}+x^{2}\mathbf{j}\) along the upper half of the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) from \((a,0)\) to \((-a,0)\)

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\(\mathbf{F}=3x\mathbf{i}+2y\mathbf{j}\)

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\(\mathbf{F}=2xy\mathbf{i}+xy^{2}\mathbf{j}\)

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\(\mathbf{F}=e^{x}y\mathbf{i}+e^{x}\mathbf{j}\)

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\(\mathbf{F}=e^{x}\mathbf{i}+5y\mathbf{j}\)

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Verify that the force field \(\mathbf{F}=y\mathbf{i} -x\mathbf{j}\) is not a conservative vector field. Show that the integral \(\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}\) is dependent on the path of integration by taking two paths in which the starting point is the origin \((0,0)\) and the endpoint is \((1,1)\). For one path, take the line \(x=y\). For the other path, take the \(x\)-axis out to the point \((1,0)\) and then the line \(x=1\) up to the point \((1,1)\).

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Work Find the work done by a variable force \(\mathbf{F} (x,y) =( x^{2}\mathbf{i}+y\mathbf{j})\) to move an object along \(C\) traced out by the parametric equations \(x=t\), \(y=t^{2},\) \(0\leq t\leq 3.\)

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Work Find the work done by a variable force \(\mathbf{F} (x,y) =\left( \cos x\mathbf{i}+\sin y\mathbf{j}\right)\) to move an object along the curve \(C\) traced out by the parametric equations \(x=t\), \(y=\sqrt{t},\) \(0\leq t\leq \pi\).

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Work An object moves in a clockwise direction from the origin to the point \((2a,0)\) along the upper half of the circle \((x-a)^{2}+y^{2}=a^{2}\) and is acted upon by a force \(\mathbf{F}\) with constant magnitude \(3\) and with direction \(\mathbf{i}+\mathbf{j}\). Find the work \(W\) done by this force.

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Work An object moves along the curve \(\mathbf{r}(t)=64\sqrt{3}t\,\mathbf{i}+(64t-16t^{2})\mathbf{j}\) from \(t=0\) to \(t=4\) and is acted upon by a force \(\mathbf{F}\) whose magnitude is directly proportional to the speed of the object and whose direction is opposite to that of the velocity. Find the work \(W\) done by this force.

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Minimizing Work An object moves from the point \((0,0)\) to the point \((1,0)\) along the curve \(y=ax(1-x)\) and is acted upon by a force given by \(\mathbf{F}(x,y)=(y^{2}+1)\mathbf{i}+(x+y)\mathbf{j}\). Find \(a\) so that the work \(W\) done is a minimum.

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Molecular Work The repelling force between a charged particle \(P\) at the origin and an oppositely charged particle \(Q\) at \((x,y)\) is \[ \mathbf{F}(x,y)=\dfrac{x}{(x^{2}+y^{2})^{3/2}}\mathbf{i}+\dfrac{y}{ (x^{2}+y^{2})^{3/2}}\mathbf{j} \]

Find the work done by \(\mathbf{F}\) as \(Q\) moves along the line segment from \((1,0)\) to \((-1,2)\).

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Work Find the work \(W\) done by the force field \(\mathbf{F} ( x,y,z) =x\mathbf{i}+y\mathbf{j}+\mathbf{k}\) in moving an object along the helix \(\mathbf{r}=\mathbf{r}(t) =\sin t\,\mathbf{i}+\cos t\mathbf{j}+t\mathbf{k}\), \(0\leq t\leq \pi\).

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Work Find the work \(W\) done by the force field \(\mathbf{F} ( x,y,z) =x^{2}\mathbf{i}+y^{2}\mathbf{j}+z^{2}\mathbf{k}\) in moving an object along the helix \(\mathbf{r}=\mathbf{r}(t)=2\cos t\,\mathbf{i}+2\sin t\mathbf{j}+2t\mathbf{k}\), \(0\leq t\leq \pi\).

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Work Suppose sea level is the \(xy\)-plane and the \(z\)-axis points upward. Then the gravitational force on an object of mass \(m\) near sea level may be taken to be \(\mathbf{F}=mg\mathbf{k}\), where \(g\) is the acceleration due to gravity. Show that the work done by gravity on an object moving from \((x_{1},y_{1},z_{1})\) to \((x_{2},y_{2},z_{2})\) along any path is \(W=mg(z_{2}-z_{1})\).

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Work

1. Suppose an object travels through a force field along a path that is always normal to the force. Show that the work done in moving the object from one point to another is \(0\).
2. Why does it follow that the gravitational field of the earth does no work on a satellite in circular orbit?

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Work Show that if the work done by a force \(\mathbf{F}(x,y)\) in moving an object along any piecewise-smooth, closed curve in an open set \(S\) in the plane is \(0\), then the work done by \(\mathbf{F}\) in moving an object from a point \(A\) to a point \(B\) in \(S\) is independent of the piecewise-smooth path chosen.

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Escape Speed

1. Show that the minimum speed needed to escape from the surface of a planet of mass \(M\) in kilograms and radius \(R\) in meters is \(v_{esc}=\sqrt{\dfrac{2GM}{R}}\), where \(G=6.67\times 10^{-11} {\rm N} {\rm m}^{2}/ {\rm kg}^{2}\).

   This speed is called the escape speed for the planet.

   (Hint: Use the relationship between kinetic energy \(K\) and the work \(W\) done by a force \(\mathbf{F}\).)

2. Find the escape speed for Earth (in kilometers per second) using \(M_{\rm Earth}=5.97\times 10^{24} {\rm kg}\) and \(R_{\rm Earth}=6.38\times 10^{6} {\rm m}\).
3. Find the escape speed for Mars (in kilometers per second) using \(M_{\rm Mars}=6.42\times 10^{23} {\rm kg}\) and \(R_{\rm Mars}=3.40\times 10^{6}{\rm m}\).
4. Find the escape speed for the Sun (in kilometers per second) using \(M_{\rm Sun}=1.99\times 10^{30} {\rm kg}\) and \(R_{\rm Sun}=6.96\times 10^{8} {\rm m}\).