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(1) A 100 kg roller coaster cart starts from rest atop a 60 meter tall hill and rolls down the slope at an angle of 45°. (a) Assuming no friction, what are the (i) potential energy, (ii) kinetic energy, and (iii) total energy of the cart when it is halfway down the track? (b) What is the speed of the cart at the bottom of the hill, again assuming that there is no friction? (c) Now we are going to assume a retarding force of 120 N is opposing the motion of the cart as shown. What is the speed of the cart at the bottom of the track in this case? (2) Consider any system governed by the equation: Ei + Wext = Ef If the final energy is plotted on the y-axis and the initial energy is plotted on the x-axis, then answer the following questions: (a) What is the theoretically expected value of the slope? What units does the slope have? (b) What is the theoretical meaning of the intercept? What units does the intercept have? (c) If the intercept is positive, what happened? If it is negative, what happened? (3) Now consider a system where an object's total kinetic energy includes rotational energy, specifically the case of a sphere. Ktot = Ktranslational + Krotational = 1/2mv^2 + 1/2I?^2 (a) Using the definition of the moment of inertia for a sphere and the angular rotational frequency in terms of mass, radius, and translational velocity, simplify the above equation to: Ktot = aKtranslational where a is a pure numerical factor. What is a? (b) Now consider questions similar to what you answered in (2), only this time we will consider a plot of final energy, all translational kinetic energy (Ktranslational) on the y-axis and the initial energy (all potential) on the x-axis. Write out the equation for Ktranslational in this case and identify the theoretically expected slope. What units does the slope have? (c) What is the theoretical meaning of the intercept? What units does the intercept have?

          (1) A 100 kg roller coaster cart starts from rest atop a 60 meter tall hill and rolls down the slope at an angle of 45°. (a) Assuming no friction, what are the (i) potential energy, (ii) kinetic energy, and (iii) total energy of the cart when it is halfway down the track? (b) What is the speed of the cart at the bottom of the hill, again assuming that there is no friction? (c) Now we are going to assume a retarding force of 120 N is opposing the motion of the cart as shown. What is the speed of the cart at the bottom of the track in this case? (2) Consider any system governed by the equation: Ei + Wext = Ef If the final energy is plotted on the y-axis and the initial energy is plotted on the x-axis, then answer the following questions: (a) What is the theoretically expected value of the slope? What units does the slope have? (b) What is the theoretical meaning of the intercept? What units does the intercept have? (c) If the intercept is positive, what happened? If it is negative, what happened? (3) Now consider a system where an object's total kinetic energy includes rotational energy, specifically the case of a sphere. Ktot = Ktranslational + Krotational = 1/2mv^2 + 1/2I?^2 (a) Using the definition of the moment of inertia for a sphere and the angular rotational frequency in terms of mass, radius, and translational velocity, simplify the above equation to: Ktot = aKtranslational where a is a pure numerical factor. What is a? (b) Now consider questions similar to what you answered in (2), only this time we will consider a plot of final energy, all translational kinetic energy (Ktranslational) on the y-axis and the initial energy (all potential) on the x-axis. Write out the equation for Ktranslational in this case and identify the theoretically expected slope. What units does the slope have? (c) What is the theoretical meaning of the intercept? What units does the intercept have?

(1) A 100 kg roller coaster cart starts from rest atop a 60 meter tall hill and rolls down the slope at an angle of 45°. (a) Assuming no friction, what are the (i) potential energy, (ii) kinetic energy, and (iii) total energy of the cart when it is halfway down the track? (b) What is the speed of the cart at the bottom of the hill, again assuming that there is no friction? (c) Now we are going to assume a retarding force of 120 N is opposing the motion of the cart as shown. What is the speed of the cart at the bottom of the track in this case? (2) Consider any system governed by the equation: Ei + Wext = Ef If the final energy is plotted on the y-axis and the initial energy is plotted on the x-axis, then answer the following questions: (a) What is the theoretically expected value of the slope? What units does the slope have? (b) What is the theoretical meaning of the intercept? What units does the intercept have? (c) If the intercept is positive, what happened? If it is negative, what happened? (3) Now consider a system where an object's total kinetic energy includes rotational energy, specifically the case of a sphere. Ktot = Ktranslational + Krotational = 1/2mv^2 + 1/2I?^2 (a) Using the definition of the moment of inertia for a sphere and the angular rotational frequency in terms of mass, radius, and translational velocity, simplify the above equation to: Ktot = aKtranslational where a is a pure numerical factor. What is a? (b) Now consider questions similar to what you answered in (2), only this time we will consider a plot of final energy, all translational kinetic energy (Ktranslational) on the y-axis and the initial energy (all potential) on the x-axis. Write out the equation for Ktranslational in this case and identify the theoretically expected slope. What units does the slope have? (c) What is the theoretical meaning of the intercept? What units does the intercept have?

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