R7_1 2/2 points (graded) Let's consider the example above again: A particle moves along the x-axis under the action of a force given by F. = - ax2, but now a = 4.0N/m2. The potential energy at & = 0m is U (0) = 0.5J. (a) What is the difference in its potential energy as it moves from @ A = 1.0m to æg = 2.0m? Give your answer in joules to three significant figures without units. 9.33 V 9.33 (b) What is the particle's potential energy at x=1.0m with respect to a given 0.5 J of potential energy at x=0 ? Give your answer in joules to three significant figures without units. 1.83 V 1.83 Types of Potential Energy Let's look at some specific examples of types of potential energy discussed last week. First, we consider each of these forces when acting separately, and then when both act together. Gravitational potential energy near Earth's surface The system of interest consists of our planet, Earth, and one or more particles near its surface (or bodies small enough to be considered as particles, compared to Earth). The gravitational force on each particle (or body) is just its weight mg near the surface of Earth, acting vertically down. According to Newton's third law, each particle exerts a force on Earth of equal magnitude but in the opposite direction. Newton's second law tells us that the magnitude of the acceleration produced by each of these forces on Earth is mg divided by Earth's mass. Since the ratio of the mass of any ordinary object to the mass of Earth is vanishingly small, the motion of Earth can be completely neglected. Therefore, we consider this system to be a group of single-particle systems, subject to the uniform gravitational force of Earth. The work done on a body by Earth's uniform gravitational force, near its surface, depends on the mass of the body, the acceleration due to gravity, and the difference in height traversed by the body. By definition, this work is the negative of the difference in the gravitational potential energy, so that difference is AUgrav = - Wgrav, AB = mg (y8 - VA) . You can see from this that the gravitational potential energy function, near Earth's surface, is U (y) = mgy+const. You can choose the value of the constant; however, for solving most problems, the most convenient constant to choose is zero for when y = 0, which is the lowest vertical position in the problem. Gravitational Potential Energy of a Hiker The summit of Great Blue Hill in Milton, MA, is 147 m above its base and has an elevation above sea level of 195 m (see picture). (Its Native American name, Massachusett, was adopted by settlers for naming the Bay Colony and state near its location.) A 75-kg hiker ascends from the base to the summit. What is the gravitational potential energy of the hiker-Earth system with respect to zero gravitational potential energy at base height, when the