Gravitational potential dimensional formula

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_University_Physics_(OpenStax)%2FBook%253A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)%2F08%253A_Potential_Energy_and_Conservation_of_Energy%2F8.02%253A_Potential_Energy_of_a_System Expand/collapse global hierarchy • Home • Bookshelves • University Physics • Book: University Physics (OpenStax) • University Physics I - Mechanics, Sound, Oscillations, and Waves (OpenStax) • 8: Potential Energy and Conservation of Energy • 8.2: Potential Energy of a System Expand/collapse global location \( \newcommand\) • • • • • • • • • • • • • • • • • • • • Learning Objectives • Relate the difference of potential energy to work done on a particle for a system without friction or air drag • Explain the meaning of the zero of the potential energy function for a system • Calculate and apply the gravitational potential energy for an object near Earth’s surface and the elastic potential energy of a mass-spring system In potential energy. We consider various properties and types of potential energy in the following subsections. Potential Energy Basics In As the football falls toward Earth, the work done on the football is now positive, because the displacement and the gravitational force both point vertically downward. The ball also speeds up, which indicates an increase in kinetic energy. Therefore, energy is converted from gravitati...

Technically, the definition of frequency can be given as the number of waves that pass through a fixed point in the unit time. The unit through which we measure frequency is hertz, and one hertz can be defined as one cycle per second. It can be generally expressed as the reciprocal of the time interval or period. The wavenumber is also sometimes used as a unit of frequency specially in spectroscopy. Frequency of the periodic motion A periodic motion repeats after a fixed time interval. If the motion repeats itself for a long time, it has a frequency. So, the frequency is the number of times the motion repeats itself in unit time. You can say that one complete motion is equal to one frequency; the number of times one complete motion is repeated in a given period is its frequency. The letter f depicts the frequency of the periodic motion. ‘Hertz’ or Hz is the unit used to measure the frequency of a periodic motion. Frequency (f) = 1/T. This formula is explained in detail in the next part. The formula of a frequency of periodic motion The time period, T, is the time required by the motion to repeat itself. The standard unit to measure the time period is seconds. Frequency (f) is the number of times a motion repeats itself in unit time. Hertz (Hz) is used to measure the frequency. There exists an inverse relationship between time period and frequency. Mathematically, it can be depicted as follows: f = 1/T Equation (1) is the formula of periodic motion. Dimensional Formula of F...

• Explain gravitational potential energy in terms of work done against gravity. • Show that the gravitational potential energy of an object of mass at height on Earth is given by • Show how knowledge of the potential energy as a function of position can be used to simplify calculations and explain physical phenomena. Work Done Against Gravity Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section. Let us calculate the work done in lifting an object of mass through a height such as in The work done on the mass is then We define this to be the gravitational potential energy put into (or gained by) the object-Earth system. This energy is associated with the state of separation between two objects that attract each other by the gravitational force. For convenience, we refer to this as the gained by the object, recognizing that this is energy stored in the gravitational field of Earth. Why do we use the word “system”? Potential energy is a property of a system rather than of a single object—due to its physical position. An object’s gravitational potential is due to its position relative to the surroundings within the Earth-object system. The force applied to the object is an external force, from outside the system. Wh...

In 3+1D, I can calculate the total force due to gravity acting on a point on the surface of the unit sphere of constant density, where I choose units so that all physical constants (as well as the density of the sphere) is 1: $$F = 4\int_$? Is the "right" gravity potential in 2+1D something like $G m_1 m_2 \log r$? If so, why, and isn't it a paradox if point masses in 2+1D orbit according to a different law than co-planar point masses in 3+1D? You're right that if you take Newton's law of gravity as is and apply it to a 2D universe, you'll get an infinite result. So you do need to use a modified theory in two dimensions, or indeed in any number of dimensions other than three. The proper way to do this is using general relativity, and if you apply GR to 2+1D spacetime, you get something that looks basically nothing like gravity as we know it. In particular, space is only distorted or "curved" where there is actually mass, unlike our universe where the distortion extends beyond the region that actually contains the mass. Because that distortion is what we recognize as gravity, in a 2+1D world there would be no gravitational attraction. The presence of mass would cause some geometrical oddities, but there would be no force acting between separated masses. For details, see e.g. Before GR was invented, on the other hand, physicists would have tried to generalize Newtonian gravity to other numbers of dimensions using Gauss's law for gravitation, which is exactly equivalent to Ga...

Continuing from last time, we defined the gravitational potential (not the potential energy!) which is related to the gravitational field as \( \vec) \) - yet another way to obtain the gravitational vector field! Next time, we'll continue to see how Gauss's law can be used in practice. Powered by Hugo.

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This article is about Gauss's law concerning the gravitational field. For analogous laws concerning different fields, see In Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to The form of Gauss's law for gravity is mathematically similar to Qualitative statement of the law [ ] ∂ V ) denotes a surface integral over a closed surface, • ∂ V is any closed surface (the boundary of an arbitrary volume V), • d A is a V, and whose direction is the outward-pointing • g is the • G is the universal • M is the total mass enclosed within the surface ∂ V. The left-hand side of this equation is called the charge can be either positive or negative, while mass can only be positive. Differential form [ ] The differential form of Gauss's law for gravity states ∇ ⋅ g = − 4 π G ρ , which is the differential form of Gauss's law for gravity. It is possible to derive the integral form from the differential form using the reverse of this method. Although the two forms are equivalent, one or the other might be more convenient to use in a particular computation. Relation to Newton's law [ ] Deriving Gauss's law from Newton's law [ ] Gauss's law for gravity can be derived from g ( r ) = − G M r 2 e r where • e r is the radial • r is the radius, | r|. • M is the mass of the particle, which is assumed to be a A proof using vector calculus is shown in the box below. It is mathematically identical to the proof of Outline of proof g( r), t...

Learning Objectives By the end of this section, you will be able to: • Explain gravitational potential energy in terms of work done against gravity. • Show that the gravitational potential energy of an object of mass m m at height h h on Earth is given by PE g = mgh PE g = mgh. • Show how knowledge of the potential energy as a function of position can be used to simplify calculations and explain physical phenomena. Work Done Against Gravity Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section. Let us calculate the work done in lifting an object of mass m m through a height h h, such as in mg mg . The work done on the mass is then W = Fd = mgh W = Fd = mgh . We define this to be the gravitational potential energy ( PE g ) ( PE g ) put into (or gained by) the object-Earth system. This energy is associated with the state of separation between two objects that attract each other by the gravitational force. For convenience, we refer to this as the PE g PE g gained by the object, recognizing that this is energy stored in the gravitational field of Earth. Why do we use the word “system”? Potential energy is a property of a system rather than of a single object—due to its physical position. An object’s gravitational po...