Forces (2)

Question 1

Angular Velocity

Answer 1

• Angular displacement: The change in angle, (radians), as a body rotates around a circle.

• Angular Velocity (⍵): The rate of change of angular displacement over time, where the velocity changes direction but speed remains constant in uniform circular motion.

• Key relationships:
  
  • Greater rotation angle (θ) in a given time results in higher angular velocity (⍵).
  • Objects farther from the center (larger r) have smaller angular velocity due to less frequent directional changes

Question 2

Frequency and period in a circle

Answer 2

• Frequency (f): The number of complete revolutions per second, measured in hertz (Hz) or s⁻¹.

• Period (T): The time taken for one complete revolution, measured in seconds (s).

Question 3

Centripetal Acceleration

Answer 3

• The acceleration of an object directed towards the centre of a circle when it moves in circular motion at a constant speed.
• Perpendicular to the object’s velocity.

Question 4

Centripetal Force

Answer 4

• The resultant force directed towards the centre of a circle, necessary to maintain uniform circular motion.

Question 5

Linear Velocity in Circular Motion

Answer 5

Uniform Circular Motion:

• Linear velocity (v): Depends on linear displacement, not angular displacement.
• As the radius (r) increases, linear velocity (v) increases.

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Angular vs. Linear Speed:

• Angular speed and angular velocity are independent of the radius.
• Linear speed depends directly on the radius of the circle.

Question 6

Centripetal Force examples

Answer 6

-Friction between car tyres and road
-tension in rope
-gravity

Question 7

Simple Harmonic Motion

Answer 7

• Definition: SHM is a type of oscillation where acceleration is proportional to displacement but acts in the opposite direction

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• Defining Equation: a ∝ −x, where a is acceleration and x is displacement.

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• Conditions for SHM:
  
  • Acceleration is proportional to displacement.
  • Acceleration acts in the opposite direction to displacement.

Question 8

SHM graphs

Answer 8

Undamped SHM Graphs:

• Periodic and described using sine and cosine curves.

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Displacement-Time Graph:

• Amplitude (A): Maximum displacement (x).
• Time period (T): Time for one full cycle.
• May not start at 0; maximum displacement indicates amplitude.

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Velocity-Time Graph:

• 90° out of phase with the displacement-time graph.
• Velocity is the gradient of the displacement-time graph.
• Maximum velocity occurs at equilibrium (displacement = 0).

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Acceleration-Time Graph:

• Reflection of the displacement graph along the x-axis.
• 90° out of phase with the velocity-time graph.
• Acceleration is the gradient of the velocity-time graph.
• Maximum acceleration occurs at maximum displacement.

Question 9

Isochronous oscillation

Answer 9

The period of oscillation is independent of the amplitude

Question 10

Conditions for SHM

Answer 10

1. Isochronous oscillations: The time period is independent of amplitude.
2. Central equilibrium point: The object oscillates around this point.
3. Continuous variation: Displacement, velocity, and acceleration change sinusoidally over time.
4. Restoring force: Always directed towards the equilibrium point.
5. Hooke’s Law: The magnitude of the restoring force is proportional to the displacement from equilibrium.

Question 11

Potential and Kinetic energy in SHM

Answer 11

Energy in Simple Harmonic Motion (SHM):

• An object exchanges kinetic energy (KE) and potential energy (PE) as it oscillates.
• Potential energy depends on the restoring force (e.g., gravitational or elastic).

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Energy at Key Points:

• Equilibrium:
  
  • PE = 0, KE = maximum (velocity is maximum).
• Maximum displacement (amplitude):
  
  • PE = maximum, KE = 0 (velocity is zero).

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Total Mechanical Energy:

• Constant throughout the motion (assuming no damping).
  E = 1/2 m ω² x₀²

Question 12

Time period in a spring/pendulum

Answer 12

• T = 2π √(m÷k)
• T = 2π √(l÷g)

Question 13

Free oscillations

Answer 13

• Occur when a system is displaced and oscillates without energy transfer to or from the surroundings.
• Oscillate at the natural (resonant) frequency.
• Only internal forces act; no external forces or energy input.
• Amplitude remains constant in ideal conditions (e.g., in a vacuum).
• Energy alternates between kinetic (Ek) and potential energy (Ep) without loss.

Question 14

Forced Oscillations

Answer 14

• Forced oscillations occur when a periodic external force is applied to sustain motion.
• The external force does work against damping forces (like friction), replacing lost energy.
• The system oscillates at the frequency of the external driving force, not its natural frequency.

Question 15

Natural frequency

Answer 15

Frequency of an oscillation when it is oscillating freely

Question 16

Resonance

Answer 16

Resonance:

• Occurs when the driving frequency matches the natural frequency of an oscillating system.
• Results in a significant increase in amplitude.

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Key Points:

• Energy transfer from the driving force to the system is most efficient at resonance.
• The system achieves maximum kinetic energy and vibrates with maximum amplitude.

Question 17

Damping

Answer 17

Damping:

• The reduction of energy and amplitude caused by resistive forces (e.g., friction, air resistance).
• Energy is proportional to amplitude².

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Damped Oscillations:

• Amplitude decreases over time, and the system eventually comes to rest at the equilibrium position.
• The frequency of oscillations remains constant as the amplitude decreases.

Question 18

Types of damping

Answer 18

Light Damping:

• Amplitude decreases exponentially over time.
• Oscillations continue with gradually decreasing amplitude.
• Frequency and time period remain constant.

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Critical Damping:

• The oscillator returns to equilibrium without oscillating in the shortest time possible.
• Displacement-time graph: No oscillations, fast decrease to zero.
• Example: Car suspension systems.

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Heavy Damping:

• The oscillator returns to equilibrium slowly without oscillating.
• Displacement-time graph: Slow decrease in amplitude.
• Example: Door dampers preventing doors from slamming.

Question 19

Amplitude-Frequency Graphs

Answer 19

Resonance Curve:

• A graph of driving frequency (f) vs. amplitude (A) of oscillations.

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Key Points:

1. f < f₀: Amplitude increases as driving frequency approaches the natural frequency (f₀).
2. f = f₀ (resonance): Amplitude reaches its maximum.
3. f > f₀: Amplitude decreases as driving frequency exceeds the natural frequency.

Question 20

The Effects of Damping on Resonance

Answer 20

Effect of Damping on Resonance:

• Damping reduces the amplitude of resonance vibrations.
• The natural frequency (f₀) remains unchanged.

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Changes with Increased Damping:

• Amplitude of resonance decreases, lowering the peak of the resonance curve.
• The resonance peak broadens.
• The peak shifts slightly to the left of the natural frequency when heavily damped.

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Conclusion:

• Damping reduces the sharpness of resonance and lowers the amplitude at the resonant frequency.

Question 21

Examples of Forced Oscillations & Resonance

Answer 21

Microwaves and Resonance:

• Microwaves use electromagnetic radiation at a frequency that resonates with the natural frequency of water molecules in food.
• Resonance causes water molecules to oscillate, generating heat energy through molecular friction.

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Heating Mechanism:

• Unlike conduction or convection, microwaves transfer heat directly via radiation to water molecules.
• Resonance ensures efficient energy transfer, as the microwave frequency matches the natural frequency of water molecules for optimal energy absorption.

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Efficiency:

• The principle of forced oscillations and resonance allows microwaves to heat food more efficiently than traditional methods.

Question 22

force field

Answer 22

Region of space where an object will experience a force

Question 23

Gravitational Fields

Answer 23

• A gravitational field is a force field generated by any object with mass, causing other masses to experience an attractive force.
• Gravitational fields are created around any object with mass and influence other nearby masses.
• They represent the action of gravitational forces acting between masses in their vicinity.

Question 24

Gravitational field lines

Answer 24

• Show the direction of the gravitational force on a mass, always pointing toward the center of mass due to gravity’s attractive nature.
• Around a point mass, the field is radial, with lines converging toward the center.
• Represent the direction of acceleration for a mass placed in the field.

Question 25

Radial Vs Uniform field lines

Answer 25

Radial Fields:

• Non-uniform, with field strength (g) varying with distance from the center.
• Represented by radially inward lines.

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Uniform Fields:

• Have equally spaced parallel lines, with field strength (g) remaining constant.
• On the Earth’s surface, gravitational field lines are parallel, approximating a uniform field.

Question 26

Newton’s Law of Gravitation

Answer 26

Newton’s Law of Gravitation:

• The gravitational force (F) is:
  
  • Directly proportional to the product of two masses (m₁ and m₂).
  • Inversely proportional to the square of their separation (r).
• The negative sign indicates the force is attractive.

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Distance (r):

• Measured between the centers of the two masses.
• For objects above a planet’s surface, add the planet’s radius to the height above the surface.

Question 27

Gravitational field strength in radial

Answer 27

Gravitational Field Strength (g) in a Radial Field:

• Equation: g = -GM/r²,
  where:
  
  • G = gravitational constant.
  • M = mass of the object.
  • r = distance from the center of the mass.
• The negative sign indicates the field is attractive, with field lines pointing toward the mass.

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Key Points:

• g decreases with increasing r, following an inverse square law.
• On Earth’s surface, g ≈ 9.81 N kg⁻¹, but outside Earth, g is not constant.

Question 28

GPE in a radial field

Answer 28

Gravitational Potential:

• Defined as the work done per unit mass to bring a mass from infinity to a defined point.
• Always negative because work is required to move mass away from the gravitational source (attractive).
• Infinity is the reference point where gravitational potential is zero.
• Potential becomes less negative as mass is moved away from the source.

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Gravitational Potential Energy (G.P.E.):

• The energy an object has due to its position in a gravitational field.

Question 29

Gravitational potential difference

Answer 29

• Gravitational potential difference (ΔV) is the difference in gravitational potential between two points at different distances from a mass.
• ΔV is given by ΔV = Vf – Vi, where Vf is the final potential and Vi is the initial potential.

Question 30

Escape velocity

Answer 30

Escape Velocity:

• The minimum speed required for an object to escape a gravitational field without further energy input.
• Depends on the mass (M) and radius (r) of the object creating the gravitational field.
• Same for all objects in the same gravitational field.

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Equation:

• v = √(2GM/r), where:
  
  • v = escape velocity.
  • G = gravitational constant.
  • M = mass of the object.
  • r = distance from the object’s center.

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Key Point:

• An object reaches escape velocity when all its kinetic energy is transferred to gravitational potential energy.

Question 31

Centripetal Force on a Planet

Answer 31

Satellite Motion:

• The gravitational force on a satellite acts as the centripetal force, always perpendicular to the direction of travel.

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Orbital Speed Equation:

• Equating gravitational force to centripetal force gives:
  v = √(GM/r), where:
  
  • v = orbital speed.
  • G = gravitational constant.
  • M = mass of the central object.
  • r = distance from the center of the object to the satellite.

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Key Point:

• The mass of the satellite (m) cancels out, showing that orbital speed is independent of the satellite’s mass.

Question 32

Orbital period

Answer 32

• The orbital speed 𝑣 of a planet or satellite in a circular orbit is given by the circular motion formula v = 2π/T
• v² = GM/R
• Equal the two and solve for T²

Question 33

Force Distance Graphs for gravitational force

Answer 33

Force-Distance Graph for Gravitational Fields:

• The graph is a curve because force (F) is inversely proportional to r².
• The area under the graph represents work done or energy transferred.

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Gravitational Potential Energy:

• When a mass m moves further from mass M, its gravitational potential energy increases, requiring work to be done.
• The area between two points on the graph gives the change in gravitational potential energy of mass m.

Question 34

Geostationary satellite

Answer 34

Geostationary Orbit:

• A satellite remains directly above the equator, in the plane of the equator, and always at the same point above Earth’s surface.
• The satellite moves west to east with an orbital period of 24 hours, matching Earth’s rotation.

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Applications:

• Used for telecommunication transmissions and television broadcasts, where the satellite amplifies and sends signals back to Earth.
• Receiver dishes on Earth can be fixed, as they always point to the same spot in the sky due to the satellite’s fixed position.

Question 35

Kepler’s Three Laws of Motion

Answer 35

Kepler’s First Law:

• The orbit of a planet is an ellipse with the Sun at one of the two foci.
• Orbits can be circular or highly elliptical.

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Kepler’s Second Law:

• A line joining a planet and the Sun sweeps out equal areas in equal time intervals.
• Planets move faster when closer to the Sun and slower when farther away.

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Kepler’s Third Law:

• The square of the orbital period (T) is directly proportional to the cube of the orbital radius (r):
  T² ∝ r³
• A constant (k) applies to all planets in the same system.

Question 36

Circular Motion on a Banked Track/Vertical Loop

Answer 36

Circular Motion in a Vertical Loop:

• Bottom of the Loop:
  
  • Centripetal Force (F) = R - mg (difference between upward force R and weight mg).
• Top of the Loop:
  
  • Centripetal Force (F) = R + mg (sum of upward force R and weight mg).
• Side of the Loop:
  
  • Centripetal Force (F) = R (equal to the upward force).

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Circular Motion on a Banked Track:

• Forces:
  
  • Weight (mg): Acts vertically downward.
  • Normal Force (N): Acts perpendicular to the surface.
• Components of Normal Force (N):
  
  • Vertical: Balances weight.
  • Horizontal: Provides centripetal force.
• Angle of Banking (θ): Determines the ratio of the normal force’s components, reducing reliance on friction.

Question 37

SHM in a pendulum

Answer 37

Height Difference in Pendulum Motion:

• The height difference is not the amplitude.
• Use mgh = KE to relate height to kinetic energy.
• Maximum velocity (vₘₐₓ) can be found using the energy equation.

Question 38

Weight on equator Vs weight on north pole

Answer 38

Less on equator because some of the weight is used as centripetal force