Mama Physics 101 PDF

# Gravitation and Energy ## Escape Speed ### What is it? This is the minimum speed an object needs to escape from the gravitational influence of a massive body, like a planet or a star without additional propulsion. This is the speed of a rocket when it escapes from the gravitational influence of...

# Gravitation and Energy ## Escape Speed ### What is it? This is the minimum speed an object needs to escape from the gravitational influence of a massive body, like a planet or a star without additional propulsion. This is the speed of a rocket when it escapes from the gravitational influence of the Earth. Or the speed of astronauts leaving the Earth to the moon. - The escape speed depends only on the mass of the massive object, *M*, from which the projected object (eg rocket) escapes. - If a projectile is fired upwards, it will slow down as it moves up; stops momentarily and come down to Earth. There is a certain minimum initial speed that will cause a projectile to move upward forever. - The initial minimum speed is the escape speed, *V*. ### How to calculate it Consider a mass, *m*, leaving the surface of a planet for some other astronomical place body with the escape velocity *V*. Consider a rocket of mass *m* leaving the Earth of mass *M*, with a minimum speed *V*. - The kinetic energy of the rocket is: ***KE = 1/2*mV²*** - The potential energy *M* *m* is: ***-GMm/R*** - Where *R* is the radius of the planet, Earth in this case. The total mechanical energy of the system is: *E = KE + PE* *E = 1/2*mV² + (-GMm/R)* *E = 1/2*mV² + GMm/R = 0; Since E = 0 at infinity* Therefore: *1/2*mV² = GMm/R* *mv² = 2GMm/R* *V² =2GM/R* *V = √(2GM/R) ms⁻¹* This is the escape speed of the rocket from the gravitational influence of the Earth. This is valid for all astronomical cases - for a body to escape the gravitational influence (pull) of an astronomical body (natural or artificial Satellites) of mass *M* and radius *R*. It is seen that V = √(2GM/R) depends only on the mass of the body *M*, from which the projectile escapes not on the mass of the escaping body, *m*. *V* can therefore be calculated if *M* and *R* are known. For the Earth, *V* is about 11.2 km/s or 25000 mph. For a projectile to escape from the gravitational influence of the Earth: *V < √(2GM/R)*, the projected body will not escape. Check: - *M* = Mass of the Earth = 5.98 x 10^24 kg - *R* = Radius of the Earth = 6. 37 x 10^6 m The escape speed from the gravitational influence of the Earth, *V* = √(2GM/R). *V = √(2 x 6.67 x 10⁻¹¹ x 5.98 x 10^24 / 6.37 x 10^6)* *V = 11.2 km s⁻¹* *** # Gravitation and Energy ## Gravity Gravity is a fundamental force that attracts two masses towards each other. Gravity causes things to fall to the ground. Gravity causes planets to move around the Sun and other planets and stars. ## Law of Universal Gravitation The law states that all particles in the Universe (all objects) with a force that is proportional to the product of the masses of the particles, and inversely proportional to the square of the distance between the particles. The law is expressed as: *F = Gmim2 (N)* - m1 and m2 are the masses of the particles - F is the force, the force is attractive - r is the distance between the bodies - G is the Universal Gravitational Constant = 6.67 x 10⁻¹¹ Nm² kg⁻² The law holds between the Sun and the Earth, and any other stars and its satellites. ## Relation between Gravity and Energy Objects in a gravitational field possess energy depending on their masses and height. The greater the height, the greater the potential energy. Gravity and energy interact dynamically throughout the universe affecting everything from the orbits of the planets to the bending of light around corners and massive objects. It affects the motion of electrons and the nucleus in an atom, and so generally around subatomic particles. Energy is the capacity to do work. Energy comes in the form of: - Potential energy (stored energy based on position) - Kinetic energy (related to motion) - Thermal energy - Nuclear energy, etc ## Gravitation near the surface of the Earth *F = GMm/r²*. If the particle is released it will fall towards the center of the Earth as a result of the gravitational force. *F = mg*, where g is the **gravitational acceleration**. So: *mg = GMm/r²* *g = GM/r²* for various altitudes. Assumption: Earth is not an inertial Frame Earth is rotating round the Sun and spinning on its axis. The **gravitational acceleration** is not quite the same as the free fall acceleration *g*. Why are the two acceleration not the same ie *g ≠ g* ? Three reasons are: 1. Mass of the Earth is not uniformly distributed. 2. Earth is not a perfect sphere, the Earth is an ellipsoid. 3. Earth rotates and at the same time spins on its axis. As a result of these, the weight of the body *mg* differs from the force of gravity because **mg ≠ Fm** 1. Since the Earth is not uniformly disturbed, the mass is not uniformly distorted radially, so density of the Earth varies radially. The density of the crust (conter section) varies from region to region over the Earth's surface. 2. Earth is not a sphere, Earth is approximately ellipsoid flattened at the poles and bulging at the Equator. Its equatorial radius is greater than its polar radius. This is why *g* changes as one moves from the equator towards the poles. As one moves, one gets near the core of the Earth, *r* gets smaller and *g* increases due to increase in density. 3. The Earth rotates. Because of the above, the free fall acceleration *g* near the surface of the Earth is not the same as the gravitational acceleration. The weight of a particle (W = mg) is not the same as the magnitude of the gravitational force on the particle. ## Gravitational acceleration / free fall acceleration The gravitational acceleration *a* of a particle (*m*) is due to the gravitational force acting on the particle. When the particle is at a distance *r* from the centre of a uniform spherical body of mass *M*, the magnitude *F* of the gravitational force on the particle is: *F = ma* *GMm/r² = ma* *a = GM/r²* where *GMm/r²* is the **Newton's gravitational force**. *a* = gravitational acceleration. ### Free fall acceleration - The mass of the Earth is not distributed uniformly - The planet is not perfectly symmetrical - Earth is rotating. Rotation axis is through north and south poles The rotating body experiences a centripetal force and hence centripetal acceleration *a*. *a* is directed towards the centre of the Earth. If *w* = ω is the angular velocity, then *a = w²r = ω²r m s⁻²* So centripetal force = m * w *r = mv²/R where *R* is the radius of the Earth for a body orbiting round the Earth. Let *FN* be a normal force acting on the body. Therefore: *FN - mg = mw²R = mv²/R* *mg = due to gravitational effect* *FN = mg = weight of the body* *mg > mg - mw²R* *g2 = g1 - w²R* *g2 = g1 - w²R = g1 - mv²/R* - Free fall acceleration - Gravitational acceleration - Centripetal acceleration Free fall acceleration is less than the gravitational acceleration. *g > g2* **Which is Greater** At the equator, gravitational acceleration is greatest at the equator, *g1 - g2 = w²R* ie the difference between the gravitational acceleration, because the radius of the Earth is greatest at the equator. The radius *R = 6.37 x 10^6 m* (radius of the earth). *g1 - g2 = 0.034 ms⁻²* which can be neglected in comparison with the free fall acceleration of 9.8 ms⁻². Hence the weight of a body *mg* and gravitational acceleration *g* are comparable. Gravitational acceleration *g* - the free fall acceleration *(g1- g2)* = w²R. This value is negligibly small compared with the free fall acceleration, or is *(g1- g2)* is negligibly small. ## Gravitation inside the Earth A uniform shell of matter exerts no net gravitational force on a particle located inside it. Why? - If the mass of the Earth were uniformly distributed, the gravitational force on the particle would be maximum at the Earth’s surface and would decrease as the particle moved outward away from the planet. - As the particle moved down into the Earth, the gravitational force would tend to increase because the particle would tend to move towards the center of the Earth *(g = GM/r²)* because *r* is decreasing. - It would tend to be decreasing because there would be an increasing shell of the mantle material lying outside. However, it is assumed that the mass of the Earth *M* is concentrated on a particle at the center of the Earth. The magnitude *F* of the gravitational force on this particle is: *F= GMm/r²* Assume a uniform density, ρ. Density = Inside Mass/Total Volume Density = Inside Mass/Total Volume or Inside Mass/4/3πr³ = 3M/4πR³ = ρ So: *4/3πR³ρ = 3M* *4/3πR³ *ρ = M* *M = 4π/3R³ρ = Mass of the Earth* *ρ = M/4πR³ = 3M/4πR³ = ρ* *ρ = M/4πR³ = 3M/4πR³ = 3Mms/4πr³ where Mms = mass of the earth at the centre* Then: *M = 4π/3R³ *ρ = Mmsρ³/R³* *ρ = 3M/4πR³* *Mms = 4/3πR³ρ = M* *Mms = 4π/3πR³M/4πR³ = MR³/R³* *Mms = M* ## Gravitation within a spherical shell A uniform shell of matter exerts no force on a particle located inside the spherical Shell. This means that if a particle is located inside a uniform solid sphere at a distance *r* from the centre of the solid sphere, the gravitational force exerted on the particle is due to the mass that lies inside the sphere of radius *r* (the inside sphere). The magnitude of the force is given as: *F = GMm/R³* - *M* = mass of the sphere -*R* = radius of the solid sphere - *m* = mass of the particle ## Problem 1 Calculate the escape speed of a rocket from the Earth; spacecraft 5000 kg. And determine the kinetic energy it must have at the surface of the Earth to move infinitely far away from the Earth. Calculation: *V = √(2GM/R)* *V = √(2 x 6.67 x 10⁻¹¹ x 5.97 x 10^24 / 6.3 x 10⁶)* *V = 1.12 x 10⁴ ms⁻¹* *KE = 1/2*mv²* *KE = 1/2 x 5000 x (2GM/R)* **KE = 1/2 x 5000 x (2 x 6.67 x 10⁻¹¹ x 5.97 x 10^24/ 6.3 x 10⁶)* J* **KE = 3. 3 x 10¹⁰ J** ## Energy of a Satellite in Orbit ### Mass of a body (m) Unit = (kg) Mass is the measure of inertia of a body - ie the reluctance of a body to undergo linear acceleration. Unit of mass is kg. A force *F* applied to a mass *m* to give it an acceleration *a*, is expressed as: *F = ma (N)* If a force *F* que's a body, the mass of an acceleration *a*, a kilogram the same force gives a kilogram mass a *3* times the acceleration *(=3a)* *F = m1a1* *F = m2a2* *F = m1a1* *F = m1a1 /3 = m2a2*, therefore, *3m1a1 = m1a1* *a2 = 3a1* The mass of an object is constant. Mass does not vary from place to place but weight of an object (W=mg) varies from place to place. Since *g* varies from place to place, the weight varies. Weight varies from place to place as *g* varies on various locations of the Earth. ### Weight of an object: Weight of an object is the gravitational force exerted on a body. It depends on the distance between the body and the centre of the Earth. *mg z F = GmM/r²* If *m* = 1kg; then *F z 6.67 x 10⁻¹¹ x 1 x 5.98 x 10^24 / (6. 37 x 10⁶)²* *F = 9.83N = weight of 1kg body* *F = 9.83N x 70 = 688.1 N* ### Energy of the Satelitte *Ms >> ME*, where *M* is the gravitational pull on a body by the Earth, is much greater than the gravitational pull of the Sun on the same body? - **Question** An object of mass *m* is located on the surface of a spherical planet of mass *M* and radius *R*. The escape speed from the planet does not depend on the following: - *M* - *m* - Density of the planet - *R* - Acceleration due to gravity of the planet. - **What do you think the force on a mass *m* would be at the center of the Earth?** - **Determine the order of magnitude of the gravitational force that you exert on another person 1m away.** ## Gravitation and Energy - **Gravity** Gravity is a fundamental force that attracts two masses towards each other. Gravity causes things to fall to the ground. Gravity causes planets to move around the Sun and other planets and stars. - **Law of Universal Gravitation** *F = -GM1* *m2/r²*. It is a force of attraction. *M1*, *M2* are the masses of the bodies. *r* is the distance between the bodies. - **Energy** Energy is the capacity to do work. Energy comes in the form of: - Kinetic energy (related to motion) - Potential energy (stored energy based on position eq gravitational potential energy) - Thermal energy - Chemical energy, etc. Energy can transform from one form to another, (potential energy to kinetic energy and vice versa). ## Relation between gravity and Energy Objects in a gravitational field possess potential energy depending on their masses and height. The greater the height, the greater the potential energy. Gravity and energy interact in dynamic ways throughout the universe affecting everything from the orbits of the planets to the bending of light around corners and massive objects. ## Energy Conservation in Planetary Motion Consider an object of mass *m*, moving with speed *V* in the vicinity of a massive object of mass *M* (*M>>m*). The system might be a planet moving around the Sun; a planet satellite in orbit around the Earth or a comet making a one-time flyby of the Sun. Assume the object of mass *M* is at rest in an inertial reference frame. The total mechanical energy of the two bodies system when they are separated by a distance *r* is the sum of the kinetic energy of the object of mass *m* and the potential energy of the system. *E = KE + PE = KE + U, but U = -GMm/R* So: *E = 1/2*mV² + (-GMm/R)* *E = 1/2*mV² - GMm/R* *E* may be negative, positive or zero. Value of *E* depends on the value of *V*. - *E = O* when *V = ∞* Let *FC* = centripetal force holding the object in its circular orbit. *F = mass * centripetal acceleration* *ma = mv²/R* Therefore: *GMm/r² = mv²/r* *GM/r = V²* So: *√(GM/R) = V* The Kinetic energy becomes: *1/2*mv² = 1/2 * m x GM/R = GMm/2r* We can now calculate the total kinetic energy of the satellite: *ETotal = -GMm/R + GMm/2R = -GMm/2r (circular orbit)* Total mechanical energy is negative, KE is positive and is equal to half the absolute value of the potential energy The absolute value of the total energy is also equal to the binding of the system as this amount of energy will be provided to the system to move the objects infinitely apart (to break up). The total energy is constant if the system is isolated, and a constant of motion. ## Gravitation and Energy - Gravitation Potential - Potential energy = Work done in moving a unit mass from infinity to the point in question. - *U = -GM/r* Gravitational field is equal to the negative of the **potential gradient** The -ve sign indicates that the potential falls when moving in the direction of the field. ## Escape Speed for any satellite *Vₑ = √(2GM/R)* - *M* = mass of the planet, *R* its radius - *g* = gravitational field at the surface of the planet ## Weight Weight of a body on the Earth is the gravitational force exerted on the body by the Earth. Weight of an object varies with its distance from the Earth. It is not altered by any acceleration the body may have. ## Problem 2 Calculate the gravitational potential difference between a point on the surface of the Earth and a point 1600 km above the surface of the Earth. *U1 = -GMm/R* *U2 = -GMm/(R+h)*, h = 1600km *U2 - U1 = -GMm/(R+h) - (-GMm/R)* *U2 - U1 = -GMm/(R+h) + GMm/R* *U2 - U1 = GMm(R+h)/R(R+h) + GMm(R+h)/R(R+h)* *U2 - U1 = -GMm(R)/R(R+h) + GMm(R+h)/R(R+h)* *U2 - U1 = GMm(-R + R + h)/R(R+h)* *U2 - U1 = GMmh/(R(R+h))* - Substitute values and calculate the gravitational potential difference. ## Definitions - **Define gravitational potential at a point.** The gravitational force on a body of mass *m* is: *F = GMEM/r²* - **What is the acceleration of free fall *g*?** *g = GM/R²* - **Note mg = GMm/R² ⇒ g = GM/R²** Therefore: *R = GM/g* *G1 = gR²/M* and *M = gR²/G* kg. - **Density of the Earth *ρ*** *ρ = gR²/G/Volume = gR²/G/4/3πR³* *ρ = 3gR²/G/4πR³ = 3g/4πGR kg m⁻³* Which is denser, the crust or the core of the Earth? - **If T² = 4π²R³/G, what is G?** *G = 4π²R³/T²* - **What is *R*?** *R = 3T²G/4π²* - **What is the frequency?** *f = 1/T, but T = 2π√(R³/G)* Therefore: *f = 1/2π√(G/R³)* Hertz. - **What is the angular velocity *ω* of the planet?** *ω = 2πf = 2π *1/2π√(G/R³)* rad s⁻¹ *ω = √(G/R³)* - **Which is correct?** The escape speed of a satellite from the gravitational influence of a heavier body of mass *M* is: A. (√GM/3R) B. (√2GM/R²) C. (√2GM/R) D. (√GMm/R) **Answer: C, (√2GM/R)** - **Question** Gravitational field of a point near to a mass. - **g = GM/r²** - **g = GM/r, g = GMm/r** *g* is a scalar field. True or false? *g* points towards the mass creating it. A. away; towards B. towards; parallel C. None of the above **Answer- C, None of the above** What's the reason why the gravitational acceleration is greater than the acceleration of free fall? A. The Earth moves in a circular orbit. B. The mass density of the Earth in uniform and constant value. C. The Earth is ellipsoidal D. **Answer: C, The Earth is ellipsoidal** - **Question** The weight of a body is the same as the mass. True/False? **Answer: False** Weight of one unit mass in a location where *g* is 9.8 m s⁻² is: A. Equal, same direction. B. Half and opposite. C. Half but same. D. Equal but opposite in **Answer: A, Equal, same direction** What is the force holding any? The kinetic energy of a satellite is ...to and .... - **The total energy of a satellite in space, at infinity, is:** A. *E = gravitational potential energy + kinetic energy of its motion* B. *The sum of the KE + P.E.* **Answer: A. E = gravitational potential energy + kinetic energy of its motion** The speed of escape of a rocket is: *V = √(2GM/R²)* *V = √(GMm/m)* *V = √(GMm/m)* *V = √(2GM/R²)* - **Example** A 20 kg mass is 5 m above the ground. (h = 0 on the ground). What is the gravitational potential energy? (g = 9.8 ms⁻²) Solution: *PE = mgh* *PE = 20 x 5 x 9.8 J* *PE = 980 J* This is the energy of the object due to its position in the gravitational field of the Earth. ## Energy Energy cannot be created, nor destroyed; but energy can be transformed from one form to another. In all cases the total energy in a system remains constant. ## Gravitational Potential Energy (GPE) This is the energy an object possesses because of its position in a gravitational field. If the body is assumed to be near the surface of the earth, the gravitational field is taken to be constant. The potential energy (GPE) depends on the mass of the object (*m*), the height of the object from the reference point and the strength of the gravitational field. **Potential energy (GPE) = mgh.** (Joule) - *m* = mass of the object - *g* = gravitational field strength - *h* is the height from the reference point (in metres) ## Weightlessness Weightlessness is the absence of the sensation of weight. **Zero gravity**: The sensation of having no weight. *mg = 0* but: - It occurs when an object or person is in free fall experiencing the same acceleration as the gravitational pull around it. - Weightlessness is generally associated with space travel, where astronauts appear to float because they and their spacecraft are orbiting the Earth at the same speed. ## Causes of Weightlessness: - **Free fall**: Accelerates at the same rate as gravity, eliminating the sensation of weight. - **Orbiting**: Objects in orbit around a planet are continuously falling toward the planet, but creating a sense of constant free fall. Objects in space are not weightless, they appear to be so. ## Simulated Environments Aircrafts can create brief periods of weightlessness during parabolic flight maneuvers. ## Effects - Affect human **vestibular system** leading to motion sickness - **Muscles and bones weaken** due to lack of use. - **Bone and Muscle loss** in the absence of gravity. - **Muscles lose density** - since they are not bearing weight. - **Fluid redistribution** in humans. - **Changes in vision**. - **Cardiovascular changes**. - **Immune system alterations**. ## Weightlessness - Allows for experiments that would be impossible or less precise under Earth gravity. - Near or complete absence of the sensation of weight. - Zero-g free; zero-g. - **Fluid Dynamics**: How do fluids behave under gravity, microgravity in spacecrafts and its study of blood flow. - **Crystal growth**: Proteins etc form purer crystals in microgravity - advances in drug manufacturing, and development and material sciences; microgravity, zero-gravity. - **Atomic and Quantum Physics**: Provides a stable platform to test phenomena like atomic wave interference and Einstein’s theories of fundamental physics. Improves understanding of fundamental physics. ## What can make *mg* = 0? *mg = GMm/R²* *g = GM/R²* - Put between 2 objects. - If something is 4.2 x 10^6 m in space, *g* = 8.6 m s⁻² - *g* is 88% of what is experienced on Earth. ## Free fall motion A person in a lift moves down with an acceleration *(a)* equal to the acceleration due to gravity *(g)*. When *a=g*, the motion is called a free fall. The apparent weight of the palm is *R = m(g - g) = 0*. This condition occurs in the state of weightlessness. The weight of gravity is canceled by the inertial force (centripetal force resulting from the orbit of a satellite around the earth). ## Geosynchronous Satellites - Astronauts are weightless because there is no external contact force pushing or pulling upon their bodies. - Gravity is the only force acting upon them, and because gravity acts at a distance, gravity cannot be felt - so gravity does not provide any sense of weight on their bodies. Can mass be zero? No. Mass is an intrinsic property of a body. Mass depends upon an object alone, and not on other factors. <start_of_image> Forces in a satellite *Fg = gravitational force, centripetal* *Ff = friction force, can be ignored in orbit* *Fp = propulsion force* The relationship between forces in a satellite is: *Fg = Fp - Ff* *Ff = Fp - Fg* *Fp = Fg + Ff* Force is a vector. Force is a direction. For example, *Fg = 9.8* A satellite in orbit is in a state of free fall. It is constantly falling towards the Earth. The reason it does not hit the Earth is because of its tangential velocity, which is constantly changing due to the gravitational force. This change in velocity is called centripetal acceleration. ## Gravity and the Earth - The Earth's gravitational field varies with distance. - It is stronger near the Earth and weaker further away. - The Earth's gravitational field is not uniform; it is stronger at the poles and weaker at the equator. This is because the Earth is not perfectly round, but is slightly flattened at the poles and bulging at the equator. - The Earth's gravitational field is affected by the presence of other celestial bodies. ## Applications of Gravity - Understanding the motion of celestial objects. - Developing navigation systems. - Designing spacecrafts. ## Important Notes - The gravitational acceleration at the surface of the Earth is approximately 9.8 m/s². - The escape velocity of an object is the minimum velocity required for the object to escape the gravitational pull of a planet. - The orbital velocity of a satellite is the velocity required for the satellite to maintain a circular orbit around a planet. - The period of a satellite's orbit is the time it takes for the satellite to complete one orbit around a planet.