Chapter 7

In a truck loading station at a post office, a small 0.200 \(\mathrm{kg}\) package is released from rest at point \(A\) on a track that is one-quarter of a circle with radius 1.60 \(\mathrm{m}(\mathrm{Fig} .7 .39) .\) The size of the package is much less than \(1.60 \mathrm{m},\) so the package can be treated as a particle. It slides down the track and reaches point \(B\) with a speed of 4.80 \(\mathrm{m} / \mathrm{s}\) . From point \(B,\) it slides on a level surface a distance of 3.00 \(\mathrm{m}\) to point \(C,\) where it comes to rest (a) What is the coefficient of kinetic friction on the horizontal surface? (b) How much work is done on the package by friction as it slides down the circular are from \(A\) to \(B ?\)

A truck with mass \(m\) has a brake failure while going down an ?cy mountain road of constant downward slope angle \(\alpha\) (Fig. 7.40\()\) Initially the truck is moving downhill at speed \(v_{0}\) - After careening downhill a distance \(L\) with negligible friction, the truck driver steers the runaway vehicle onto a runaway truck ramp of constant upward slope angle \(\boldsymbol{\beta}\) . The truck rump has a soft sand suffice for which the coefficient of rolling friction is \(\mu_{r}\) What is the distance that the truck moves up the rump before coming to a halt? Solve using energy methods.

A certain spring is found not to obey Hooke's law; it exerts a restoring force \(F_{x}(x)=-\alpha x-\beta x^{2}\) if it is stretched or compressed, where \(\alpha=60.0 \mathrm{N} / \mathrm{m}\) and \(\beta=18.0 \mathrm{N} / \mathrm{m}^{2} .\) The mass of the spring is negligible. (a) Calculate the potentinl-energy function \(U(x)\) for this spring. Let \(U=0\) when \(x=0\) (b) An object with mass 0.900 \(\mathrm{kg}\) on a frictionless, horizontal surface is attached to this spring, pulled a distance 1.00 \(\mathrm{m}\) to the right (the \(+x\) -direction) to stretch the spring, and released. What is the speed of the object when it is 0.50 \(\mathrm{m}\) to the right of the \(x=0\) equilibrium position?

A variable force \(\overrightarrow{\boldsymbol{k}}\) is maintained tangent to a frictionless semicircular surface (Fig. 7.41\() .\) By slow variations in the force, a block with weight \(w\) is moved, and the spring to which it is attached is stretched from position 1 to position \(2 .\) The spring has negligible mass and force constant \(k .\) The end of the spring moves in an are of radius \(a\) . Calculate the work done by the force \(\vec{F}\) .

A \(0.150-\mathrm{kg}\) block of ice is placed against a horizontal, compressed spring mounted on a horizontal tabletop that is 1.20 \(\mathrm{m}\) above the floor. The spring has force constant 1900 \(\mathrm{N} / \mathrm{m}\) and is initially compressed 0.045 \(\mathrm{m}\) . The mass of the spring is negligible. The spring is released, and the block slides along the table, goes off the edge, and travels to the floor. If there is negligible friction between the block of ice and the tabletop, what is the speed of the block of ice when it reaches the floor?

A wooden block with mass 1.50 \(\mathrm{kg}\) is placed against a compressed spring at the bottom of an incline of slope \(30.0^{\circ}\) (point \(A\) ). When the spring is released, it projects the block up the incline. At point \(B,\) a distance of 6.00 \(\mathrm{mup}\) the incline from \(A\) , the block is moving up the incline at 7.00 \(\mathrm{m} / \mathrm{s}\) and is no longer in contact with the spring. The coefficient of kinetic friction between the block and the incline is \(\mu_{\mathrm{k}}=0.50 .\) The mass of the spring is negligible. Calculate the amount of potential energy that was initially stored in the spring.

A particle with mass \(m\) is acted on by a conservative force and moves along a path given by \(x=x_{0} \cos \omega_{0} t\) and \(y=y_{0} \sin \omega_{0} t\) where \(x_{0}, y_{0},\) and \(\omega_{0}\) are constants. (a) Find the components of the force that acts on the particle. (b) Find the potential energy of the particle as a function of \(x\) and \(y .\) Take \(U=0\) when \(x=0\) and \(y=0 .\left(\text { c) Find the total energy of the particle when (i) } x=x_{0 .}\right.\) \(y=0\) and \(\left(\text { ii) } x=0, y=y_{0}\right.\)

When it is burned, 1 gallon of gasoline produces \(1.3 \times 10^{8} \mathrm{J}\) of energy. A \(1500-\mathrm{kg}\) car accelerates from rest to 37 \(\mathrm{m} / \mathrm{s}\) in 10 \(\mathrm{s}\) . The engine of this car is only 15\(\%\) efficient (which is typical), meaning that only 15\(\%\) of the energy from the combustion of the gasoline is used to accelerate the car. The rest goes into things like the internal kinetic energy of the engine parts as well as heating of the exhaust air and engine. (a) How many gallons of gasoline does this car use during the acceleration? (b) How many such accelerations will it take to burn up 1 gallon of gas?

A hydroelectric dam holds back a lake of surface area \(3.0 \times 10^{6} \mathrm{m}^{2}\) that has vertical sides below the water level. The water level in the lake is 150 \(\mathrm{m}\) above the base of the dam. When the water passes through turbines at the base of the dam, its mechanical energy is converted into electrical energy with 90\(\%\) efficiency. (a) If gravitational potential energy is taken to be zero at the base of the dam, how much energy is stored in the top meter of the water in the lake? The density of water is 1000 \(\mathrm{kg} / \mathrm{m}^{3} .(\mathrm{b})\) What volume of water must pass through the dam to produce 1000 kilo-watt-hours of electrical energy? What distance does the level of water in the lake fall when this much water passes through the dam?

Gravity in Three Dimensions. A point mass \(m_{1}\) is held in place at the origin, and another point mass \(m_{2}\) is free to move a distance \(r\) away at a point \(P\) having coordinates \(x, y,\) and \(z\) . The gravitational potential energy of these masses is found to be \(U(r)=-G m_{1} m_{2} / r,\) where \(G\) is the gravitational constant (see Exercises 7.34 and 7.35 . (a) Show that the components of the force on \(m_{2}\) due to \(m_{1}\) are $$ F_{x}=-\frac{G m_{1} m_{2} x}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} \quad F_{y}=-\frac{G m_{1} m_{2} y}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} $$ $$ F_{z}=-\frac{G m_{1} m_{2} z}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} $$ (Hint: First write \(r\) in terms of \(x, y,\) and \(z\) ) (b) Show that the magnitude of the force on \(m_{2}\) is \(F=G m_{1} m_{2} / r^{2} .\left(\text { c) Does } m_{1} \text { attract or }\right.\) repel \(m_{2} ?\) How do you know?

(a) Is the force \(\overrightarrow{\boldsymbol{F}}=\boldsymbol{C} y^{2} \hat{\boldsymbol{j}},\) where \(\boldsymbol{C}\) is a negative constant with units of \(\mathrm{N} / \mathrm{m}^{2},\) conservative or nonconservative? Justify your answer.

A cutting tool under microprocessor control has several forces acting on it. One force is \(\overrightarrow{\boldsymbol{F}}=-\alpha x y^{2} \hat{\boldsymbol{j}},\) a force in the negative \(y\) -direction whose magnitude depends on the position of the tool. The constant is \(\alpha=2.50 \mathrm{N} / \mathrm{m}^{3} .\) Consider the displacement of the tool from the origin to the point \(x=3.00 \mathrm{m}, y=3.00 \mathrm{m} .\) (a) Calculate the work done on the tool by \(\vec{F}\) if this displacement is along the straight line \(y=x\) that connects, these two points. (b) Calculate the work done on the tool by \(\vec{F}\) if the tool is first moved out along the \(x\) -axis to the point \(x=3.00 \mathrm{m}, y=0\) and then moved parallel to the \(y\) -axis to the point \(x=3.00 \mathrm{m}\) , \(y=3.00 \mathrm{m}\) . (c) Compare the work done by \(\vec{F}\) along these two paths. Is \(\vec{F}\) conservative or nonconservative? Explain.

An object has several forces acting on it. One force is \(\overrightarrow{\boldsymbol{F}}=\alpha x \hat{\imath},\) a force in the \(x\) -direction whose magnitude depends on the position of the object. (See Problem \(6.96 . )\) The constant is \(\alpha=2.00 \mathrm{N} / \mathrm{m}^{2} .\) The object moves along the following path: (1) It starts at the origin and moves along the \(y\) -axis to the point \(x=0\) , \(y=1.50 \mathrm{m} ;(2)\) it moves parallel to the \(x\) -axis to the point \(x=1.50 \mathrm{m}, y=1.50 \mathrm{m} ;(3)\) it moves parallel to the \(y\) -axis to the point \(x=1.50 \mathrm{m}, y=0 ;(4)\) it moves parallel to the \(x\) -axis back to the origin. (a) Sketch this path in the \(x y\) -plane. (b) Calculate the work done on the object by \(\overrightarrow{\boldsymbol{F}}\) for each leg of the path and for the complete round trip. (c) Is \(\overrightarrow{\boldsymbol{F}}\) conservative or nonconservative? Explain.

A Hooke's law force \(-k x\) and a constant conservative force \(F\) in the \(+x\) -direction act on an atomic ion. (a) Show that a possible potential-energy function for this combination of forces is \(U(x)=\frac{1}{2} k x^{2}-F x-F^{2} / 2 k\) . Is this the only possible function? Explain. (b) Find the stable equilibrium position. (c) Graph \(U(x)\) (in units of \(F^{2} / k )\) versus \(x\) (in units of \(F / k )\) for values of \(x\) between \(-5 F / k\) and 5\(F / k\) . (d) Are there any unstable equilibrium positions? (e) If the total energy is \(E=F^{2} | k,\) what are the maximum and minimum valnes of \(x\) that the ion reaches in its motion? If the ion has mass \(m,\) find its maximum speed if the total energy is \(E=F^{2} / k .\) For what value of \(x\) is the speed maximum?

A proton with mass \(m\) moves in one dimension. The potential-energy function is \(U(x)=\alpha / x^{2}-\beta / x,\) where \(\alpha\) and \(\beta\) are positive constants. The proton is released from rest at \(x_{0}=\alpha / \beta\) (a) Show that \(U(x)\) can be written as $$ U(x)=\frac{\alpha}{x_{0}^{2}}\left[\left(\frac{x_{0}}{x}\right)^{2}-\frac{x_{0}}{x}\right] $$ Graph \(U(x)\) . Calculate \(U\left(x_{0}\right)\) and thereby locate the point \(x_{0}\) on the graph. (b) Calculate \(v(x),\) the speed of the proton as a function of position. Graph \(v(x)\) and give a qualitative description of the motion. (c) For what value of \(x\) is the speed of the proton a maximum? What is the value of that maximum speed? (d) What is the force on the proton at the point in part (c)? (e) Let the proton be released instead at \(x_{1}=3 \alpha / \beta\) . Locate the point \(x_{1}\) on the graph of \(U(x)\) . Calculate \(v(x)\) and give a qualitative description of the motion. (f) For each release point \(\left(x=x_{0} \text { and } x=x_{1}\right),\) what are the maximum and minimum values of \(x\) reached during the motion?