Models and rules
| Module 5: Rise and the fall of the clockwork universe 5.1 Models and rules |
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| 5.1.2 | Out into Space a) Describe and explain: I) Changes of gravitational and kinetic energy II) Motion in a uniform gravitational field III) The gravitational field and potential of a point mass IV) Angular velocity in rad [math]s^{-1}[/math] V) Motion in a horizontal circle and in a circular gravitational orbit. b) Make appropriate use of: I) The terms: force, kinetic and potential energy, gravitational field, gravitational potential, equipotential surface by sketching and interpreting: II) Graphs showing gravitational potential as area under a graph of gravitational field versus distance, graphs showing changes in gravitational potential energy as area under a graph of gravitational force versus distance between two distance values III) Graphs showing force as related to the tangent of a graph of gravitational potential energy versus distance, graphs showing field strength as related to the tangent of a graph of gravitational potential versus distance IV) Diagrams of gravitational fields and the corresponding equipotential surfaces. c) Make calculations and estimates involving: I) Uniform gravitational field, gravitational potential energy change = mgh II) Energy exchange, work done, ∆E = F∆s; no work done when the force is perpendicular to the displacement, resulting in no work being done whilst moving along equi-potentials III) [math] a = \frac{v^2}{r}, \quad F = \frac{mv^2}{r} = mr\omega^2[/math] IV) The radial components: [math]F_{\text{grav}} = -\frac{GmM}{r^2}, \quad g = \frac{F_{\text{grav}}}{m} = -\frac{GM}{r^2}[/math] V) Gravitational potential energy [math]E_{\text{grav}} = -\frac{GMm}{r}[/math] VI) Gravitational Potential [math]V_{\text{grav}} = \frac{E_{\text{grav}}}{m} = -\frac{GM}{r}[/math] |
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I Changes of Gravitational and Kinetic Energy:
- When an object moves in a gravitational field, its energy changes. There are two types of energy involved:
- – Gravitational Potential Energy (GPE): The energy an object has due to its position in a gravitational field.
- – Kinetic Energy (KE): The energy an object has due to its motion.
- As an object moves in a gravitational field:
- – GPE increases as the object moves away from the center of the gravitational field.
- – GPE decreases as the object moves closer to the center of the gravitational field.
- Figure 1 Potential energy loss and gain
- – KE increases as the object gains speed.
- – KE decreases as the object loses speed.
- The total energy of the object remains constant, but the conversion between GPE and KE occurs.
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II Motion in a Uniform Gravitational Field
- A uniform gravitational field is one where the gravitational force is the same everywhere. Examples include:
- – Free Fall: An object falling towards the ground under the sole influence of gravity.
- – Projectile Motion: An object moving in a curved path under the influence of gravity.
- In a uniform gravitational field:
- – The acceleration due to gravity (g) is constant.
- – The velocity of the object changes linearly with time.
- – The position of the object changes quadratically with time.
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III. The Gravitational Field and Potential of a Point Mass
- A point mass is an object with mass concentrated at a single point. The gravitational field and potential around a point mass can be described using the following equations
- ⇒ Gravitational Field (g):
- [math]g = \frac{Gm}{r^2} [/math]
- – G: Gravitational constant
- – m: Mass of the point mass
- – r: Distance from the point mass
- Figure 2 Gravitational field
- ⇒ Gravitational Potential (V):
- [math]V = – \frac{Gm}{r}[/math]
- – V: Gravitational potential energy per unit mass
- The gravitational field and potential around a point mass:
- – Decrease with increasing distance from the point mass.
- – Are directed radially inward towards the point mass.
- Figure 3 Gravitational potential energy
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IV Angular Velocity in rad [math]s^{-1}[/math]
- Angular velocity (ω) is a measure of how fast an object rotates or revolves around a central axis. It’s typically expressed in units of radians per second (rad [math]s^{-1}[/math]).
- [math]ω = \frac{Δθ}{Δt}[/math]
- Where:
- – ω: Angular velocity (rad.[math]s^{-1}[/math] )
- – Δθ: Change in angular displacement (rad)
- – Δt: Time interval (s)
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V Motion in a Horizontal Circle:
- When an object moves in a horizontal circle, it experiences a centripetal force ([math]F_c[/math]) directed towards the center of the circle. This force is necessary to maintain the object’s circular motion.
- [math]F_c = \frac{mv^2}{r}[/math]
- Where:
- -[math]F_c[/math] : Centripetal force (N)
- – m: Mass of the object (kg)
- – v: Velocity of the object ([math]m.s^{-1}[/math])
- – r: Radius of the circle (m)
- Figure 4 An object move in a horizontal circle
- Motion in a Circular Gravitational Orbit
- When an object orbits a celestial body, such as a planet or moon, it experiences a gravitational force (directed towards the center of the celestial body. This force is necessary to maintain the object’s orbital motion.
- [math]F_g = \frac{Gm_1 m_2}{r^2} [/math]
- Where:
- -[math]F_g[/math] : Gravitational force (N)
- – G: Gravitational constant ([math][/math])
- -[math]m_1[/math] : Mass of the celestial body (kg)
- -[math]m_2[/math] : Mass of the orbiting object (kg)
- – r: Radius of the orbit (m)
- Differences:
- – Horizontal circle: Centripetal force is provided by an external agent, such as friction or tension.
- – Circular gravitational orbit: Gravitational force provides the centripetal force necessary for orbital motion.
- ⇒ Real-World Applications:
- Satellite technology: Understanding circular orbits is crucial for satellite design and operation.
- Space exploration: Gravitational orbits play a key role in space missions, such as orbiting planets or moons.
- Particle physics: Circular motion is used in particle accelerators to accelerate charged particles to high speeds.
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b) Make appropriate use of:
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I) Key Terms:
- ⇒ Force (F): A push or pull that causes an object to change its motion.
- Formula:
- [math]F = m \times a \quad \text{(Force = mass × acceleration)}[/math]
- – Unit: Newtons (N)
- ⇒ Kinetic Energy (KE): The energy an object possesses due to its motion.
- [math]KE = \frac{1}{2} m v^2 \quad \text{(Kinetic Energy = 0.5 × mass × velocity}^2\text{)}[/math]
- – Unit: Joules (J)
- ⇒ Potential Energy (PE): The energy an object possesses due to its position or configuration.
- [math]PE = mgh \quad \text{(Potential Energy = mass × gravitational acceleration × height)}[/math]
- – Unit: Joules (J)
- ⇒ Gravitational Field (g): A region around a massive object where the force of gravity acts.
- [math]g = \frac{GM}{r^2} \quad \text{(Gravitational field = \(\frac{\text{gravitational constant} \times \text{mass of object}}{\text{distance}^2}\))}[/math]
- – Unit: Newtons per kilogram (N/kg)
- ⇒ Gravitational Potential (V): The potential energy per unit mass at a point in a gravitational field.
- [math]V = -\frac{GM}{r} \quad \text{(Gravitational potential = -gravitational constant × mass of object / distance)}[/math]
- – Unit: Joules per kilogram (J/kg)
- ⇒ Equipotential Surface: A surface where the gravitational potential is constant.
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II Graphs:
- ⇒ Gravitational Potential vs. Distance:
- – The area under the graph represents the gravitational potential energy.
- Formula
- [math]\Delta V = \int (g \, dr) [/math]
- [math] \text{(Change in gravitational potential = integral of gravitational field × distance)}[/math]
- Figure 5 Gravitational potential and displacement
- ⇒ Gravitational Force vs. Distance:
- The area under the graph represents the change in gravitational potential energy.
- – Formula:
- [math]\Delta PE = \int (F \, dr) [/math]
- [math] \text{(Change in gravitational potential energy = integral of gravitational force × distance)}[/math]
- Figure 6 Gravitational force and distance
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III. Graphs:
- ⇒ Force vs. Distance (Tangent to Gravitational Potential Energy Graph):
- The force (F) is related to the tangent of the gravitational potential energy (PE) graph.
- – Formula:
- [math]F = -\frac{d(PE)}{dr} [/math]
- [math] \text{(Force = negative derivative of gravitational potential energy with respect to distance)}[/math]
- – Graph: A plot of force (F) vs. distance (r) will show the force as a function of distance.
- Figure 7 graph between Force and distance
- ⇒ Field Strength vs. Distance (Tangent to Gravitational Potential Graph):
- The field strength (g) is related to the tangent of the gravitational potential (V) graph.
- -Formula:
- [math]g = -\frac{dV}{dr} [/math]
- [math] \text{(Field strength = negative derivative of gravitational potential with respect to distance)}[/math]
- Figure 8 Field strength and distance
- – Graph: A plot of field strength (g) vs. distance (r) will show the field strength as a function of distance.
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IV Diagrams:
- ⇒ Gravitational Field Diagrams:
- Show the direction and strength of the gravitational field around a massive object.
- Can be represented by field lines or vectors.
- – Formula:
- [math]g = G \times \frac{M}{r^2} [/math]
- [math] \text{(Gravitational field strength = gravitational constant × mass of object / distance}^2\text{)}[/math]
- Figure 9 Gravitational field
- ⇒ Equipotential Surface Diagrams:
- Show the surfaces where the gravitational potential is constant.
- Can be represented by contour lines or surfaces.
- Figure 10 Equi-Potential Surface
- Formula:
- [math]V = -G \times \frac{M}{r} [/math]
- [math] \text{(Gravitational potential = -gravitational constant × mass of object / distance)}[/math]
c) Make calculations and estimates involving:
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I) Uniform Gravitational Field: Gravitational Potential Energy Change
- ⇒ Equation:
- [math] \Delta E_{\text{grav}} = mgh[/math]
- ⇒ Explanation:
- In a uniform gravitational field (where g is constant), the change in gravitational potential energy depends on the mass mmm, the gravitational field strength g, and the height difference h.
- – g: Acceleration due to gravity (approximately [math]9.8 m/s^2[/math] near Earth’s surface).
- – h: Vertical height difference between two points.
- – The gravitational potential energy increases when an object is lifted to a higher height and decreases when it falls.
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II Work Done and Energy Exchange
- Equation:
- [math]∆E = F∆s[/math]
- – F: Force applied.
- – ∆s: Displacement along the direction of the force.
- Conditions:
- – No work is done when the force is perpendicular to the displacement (e.g., when moving along an equipotential surface). Equi-potentials are regions where the gravitational potential VVV is constant.
- Explanation:
- – Work is a transfer of energy resulting from a force acting along a displacement.
- – Example: If you push a ball horizontally across an equipotential surface, its gravitational potential energy doesn’t change.
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III) Centripetal Acceleration and Force
- Equations:
- – Acceleration:
- [math]a = \frac{v^2}{r} [/math]
- – Force:
- [math]F = \frac{mv^2}{r} \\ F = mr \omega^2[/math]
- ⇒ Explanation:
- These equations describe the motion of an object in a circular path:
- – v: Linear velocity.
- – r: Radius of the circular path.
- – ω: Angular velocity [math]ω = \frac{v}{r}[/math] .
- The force acting toward the center of the circular path (centripetal force) is necessary to maintain the circular motion.
- Examples include planets orbiting stars or cars rounding a curved track.
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IV) Radial Components of Gravitational Force
- Force Due to Gravity:
- [math]F_{\text{grav}} = -\frac{GmM}{r^2}[/math]
- Gravitational Field Strength (Acceleration due to Gravity):
- [math]g = \frac{F_{\text{grav}}}{m} \\ g = -\frac{GM}{r^2} [/math]
- Explanation:
- – G: Universal gravitational constant ([math]6.674 \times 10^{-11} \, \text{N} \, \text{m}^2/\text{kg}^2[/math] ).
- – M: Mass of the larger body (e.g., a planet or star).
- – r: Distance from the center of the mass M.
- Gravitational force decreases with the square of the distance [math]r^2[/math] and acts radially inward (negative sign indicates direction toward the center).
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V) Gravitational Potential Energy
- Equation:
- [math]E_{\text{grav}} = -\frac{GMm}{r}[/math]
- Explanation:
- – Gravitational potential energy is negative because it is defined relative to infinity, where the energy is zero.
- – As the object moves closer to the mass M, its energy becomes more negative, indicating that it is bound more strongly.
- The closer the object is to the mass M, the more work is required to escape the gravitational influence.
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VI) Gravitational Potential
- Equation:
- [math]V_{\text{grav}} = \frac{E_{\text{grav}}}{m} \\
V_{\text{grav}} = -\frac{GM}{r}[/math] - Explanation:
- – Gravitational potential ([math]V_{\text{grav}}[/math] ) is the gravitational potential energy per unit mass.
- – It describes the energy required to bring a unit mass from infinity to a distance r from M.
- Equipotential Surfaces:
- – Surfaces where [math]V_{\text{grav}}[/math] is constant.
- – No work is done when moving along these surfaces.
- ⇒ Practical Applications
- Orbiting Bodies:
- – The gravitational force provides the centripetal force for planets and satellites, allowing us to use the equations for circular motion in analyzing orbits.
- Escape Velocity:
- – Using [math]E_{\text{grav}}[/math] , the escape velocity [math]v_{\text{esc}}[/math] can be derived:
- [math]v_{\text{esc}} = \sqrt{\frac{2GM}{r}}[/math]
- Energy Conservation in Gravitational Systems:
- – Total energy in a gravitational system is conserved and includes both potential and kinetic components.