Viewed Questions
Line integrals Use Green's Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. $\oint_{C} f d y-g d x,$ where $\langle f, g\rangle=\left\langle x^{2}, 2 y^{2}\right\rangle$ and $C$ is the upper half of the unit circle and the line segment $-1 \leq x \leq 1$, oriented clockwise
Vector Calculus
Green's Theorem
Area line integral In terms of the parameters $a$ and $b$, how is the value of $\oint_{C}$ ay $d x+$ bx $d y$ related to the area of the region enclosed by $C$, assuming counterclockwise orientation of $C ?$
Line integrals Use Green's Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. $\int_{c} \frac{1}{1+y^{2}} d x+y d y,$ where $C$ is the boundary of the triangle with vertices $(0,0),(1,0),$ and (1,1)
Area of regions Use a line integral on the boundary to find the area of the following regions. The region shown in the figure (FIGURE CAN'T COPY)
Questions asked
(10) [12 points] A reversible engine contains 0.20 moles of monatomic ideal gas. Initially, the gas is at \( 600 \mathrm{~K} \) and confined to \( 2.0 \mathrm{~L} \). The gas undergoes the following cycle: \( (A) \rightarrow(B) \) Isothermal expansion to \( 4.0 \mathrm{~L} \). (B) \( \rightarrow \) (C) Constant volume cooling to \( 300 \mathrm{~K} \). (C) \( \rightarrow \) (D) Isothermal compression to \( 2.0 \mathrm{~L} \). (D) \( \rightarrow \) (A) Constant volume heating to \( 600 \mathrm{~K} \). (a) Draw a PV diagram representing this cycle. (b) Which processes add heat? (c) For the processes where heat is added, calculate the heat in joules. (d) Calculate the net work in joules. (e) Calculate the efficiency of this engine. (f) Compare it to the efficiency of a Carnot engine operating between the same temperatures.
Timothy James
Numerade educator
(11) [4 points] A piston of area 90 cm slowly compresses a gas which remains at constant pressure of 15 atmospheres. The final position of the piston is 12 cm from its initial position, and the internal energy of the gas drops by 15 J. (a) Is heat removed from or added to the gas in this process? (b) Find the amount of heat and the work done by the gas.
Your solutions to (10) and (11) should include PV diagrams. (10) [6 points] To discover whether an unknown gas is monatomic or diatomic, an experimenter takes a \( 2.0 \mathrm{~L} \) sample of the gas at STP and heats this sample to \( 100^{\circ} \mathrm{C} \) at constant volume. (a) The gas absorbs \( 180 \mathrm{~J} \) of heat-energy in this process. Is this gas monatomic or diatomic? (b) The experimenter weighs the sample and finds it has a mass of \( 2.5 \mathrm{~g} \). What molecule or atom comprises the gas? (c) The experimenter then releases a piston and allows the gas to cool to room temperature \( 20^{\circ} \mathrm{C} \) under constant pressure. How much work does the gas do (take care of the sign)?
Supreeta N
Your solutions to (10) and (11) should include PV diagrams. (10) [6 points] To discover whether an unknown gas is monatomic or diatomic, an experimenter takes a 2.0 L sample of the gas at STP and heats this sample to 100°C at constant volume. (a) The gas absorbs 180 J of heat-energy in this process. Is this gas monatomic or diatomic? (b) The experimenter weighs the sample and finds it has a mass of 2.5 g. What molecule or atom comprises the gas? (c) The experimenter then releases a piston and allows the gas to cool to room temperature 20°C under constant pressure. How much work does the gas do (take care of the sign)?
(9) [10 points] You are working with a billet of copper \( 1.0 \mathrm{~cm} \times 1.0 \mathrm{~cm} \) in cross section and \( 12.5 \mathrm{~cm} \) in length at room temperature \( 20^{\circ} \mathrm{C} \). You heat it to a red-hot temperature of \( 850^{\circ} \mathrm{C} \) in an oven and then take it out and hang it from a thin insulating fiber. (a) What is the length of the hot billet? (b) How much heat energy was added to the billet? (c) Calculate the rate of energy loss of the cooling billet at \( 850^{\circ} \mathrm{C} \) and at \( 100^{\circ} \mathrm{C} \). Remember to take into account heat energy coming in from the environment at room temperature. Neglect any effects of convection. Red hot bodies have emissivity of 1 . (d) Taking half the \( 850^{\circ} \mathrm{C} \) energy loss rate as an approximate average rate of energy loss, how long does it take to cool? Hint: you are removing the energy found in part (b). Instead of waiting for the above billet to radiate the energy, you drop it into \( 75.0 \mathrm{~g} \) of water at \( 20^{\circ} \mathrm{C} \). You see some water boil away. (e) How much water is left when the billet has reached \( 100^{\circ} \mathrm{C} \) ? Hint: start with the heat energy removed from the billet to reach \( 100^{\circ} \mathrm{C} \), which is somewhat less than you calculated in the previous problem. You can look up any material parameters you need, just write them clearly and separately and cite the source.
compute Volume of solid bounded by z = 1+x^2 and z = 2-x^2
Change order of Integration of \[ \int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{2 \sqrt{x^{2}+y^{2}}}^{2} f(x, y, z) d z d y d x \text { to } d x d z d y \text { integration } \]
Melissa Munoz
compute the Volume of tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), (1,1,1)
Aishwarya Krishnakumar