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

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

Question 5

Linear Velocity

Answer 5

• velocity of an object in uniform circular motion with respect to linear displacement, not angular displacement.
• As the radius (r) of the circular path increases, the linear velocity (v) also increases.
• Angular speed and angular velocity are independent of the radius of the circle.
• 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.
• Defining Equation: a ∝ −x, where a is acceleration and x is displacement.
• 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 are periodic and can be described using sine and cosine curves.
• Displacement-time graph:
  
  • Amplitude A is the maximum value of displacement x.
  • Time period T is the time for one full cycle.
  • The graph might not start at 0; maximum displacement indicates the amplitude.
• Velocity-time graph:
  
  • It is 90° out of phase with the displacement-time graph.
  • Velocity is the rate of change of displacement and can be determined from the gradient of the displacement-time graph.
  • Maximum velocity occurs at equilibrium when displacement is zero.
• Acceleration-time graph:
  
  • It is a reflection of the displacement graph along the x-axis.
  • Acceleration is 90° out of phase with the velocity-time graph and can be determined from 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

• The oscillations are isochronous:
• There is a central equilibrium point:
• The object’s displacement, velocity, and acceleration change continuously: These quantities vary over time in a sinusoidal manner.
• There is a restoring force always directed towards the equilibrium point:
• The magnitude of the restoring force is proportional to the displacement: (Hooke’s law),( displacement from the equilibrium)

Question 11

Potential and Kinetic energy in SHM

Answer 11

• In simple harmonic motion (SHM), an object exchanges kinetic energy (KE) and potential energy (PE) as it oscillates.
• The type of potential energy depends on the restoring force (e.g., gravitational or elastic potential energy).
• At equilibrium, the object’s PE is zero and KE is maximum (velocity is also maximum).
• At maximum displacement (amplitude), PE is maximum, and KE is zero (velocity is zero).
• The total mechanical energy (sum of KE and PE) remains constant throughout the motion (assuming no damping).
• The total energy of system undergoing simple harmonic motion is defined by:
  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

• Free oscillations occur when an oscillating system is displaced and left to oscillate without any energy transfer to or from the surroundings; they oscillate at their natural (resonant) frequency.
• Only internal forces act during free oscillations, with no external forces or energy input, and the oscillations continue with a constant amplitude.
• Energy alternates between kinetic (Ek) and potential energy (Ep) without loss in ideal free oscillations (e.g., in a vacuum).

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, replacing lost energy.
• The system oscillates at the frequency of the external driving force, not its natural frequency.

Question 15

Natural frequency

Answer 15

• the frequency of an oscillation when the oscillating system is allowed to oscillate freely

Question 16

Resonance

Answer 16

• Definition: Resonance is when the frequency of an applied force equals the system’s natural frequency (or a multiple of), causing a significant increase in amplitude.
• Resonance occurs when the driving frequency matches the natural frequency of an oscillating system.
• At resonance, energy transfer from the driving force to the system is most efficient.
• The system achieves maximum kinetic energy and vibrates with maximum amplitude.

Question 17

Damping

Answer 17

• Definition: Damping is the reduction of energy and amplitude caused by resistive forces on the oscillating system
• Damping reduces the energy and amplitude of oscillations due to resistive forces like friction or air resistance.
  (** Energy is proportional to amplitude^2**)
• Damped oscillations decrease in amplitude over time and eventually come to rest at the equilibrium position.
• The frequency of damped 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.
  • The frequency and time period remain constant.
• Critical Damping:
  
  • The oscillator returns to equilibrium without oscillating in the shortest time possible.
  • The displacement-time graph shows no oscillations, with a fast decrease to zero.
  • Example: car suspension systems.
• Heavy Damping:
  
  • The oscillator returns to equilibrium slowly without oscillating.
  • The displacement-time graph shows a 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) versus amplitude (A) of oscillations.
• When f < f₀: The amplitude of oscillations increases as the driving frequency approaches the natural frequency.
• At f = f₀ (resonance): The amplitude reaches its maximum. This is the point of resonance.
• When f > f₀: The amplitude of oscillations decreases as the 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.
• Natural Frequency (f₀): The natural frequency remains the same despite damping.
• Changes with Increased Damping:
  
  • The 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.
• 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 use electromagnetic radiation at a specific frequency that resonates with the natural frequency of water molecules in food.
• Resonance causes water molecules to oscillate, generating heat energy through molecular friction.
• Unlike conventional cooking methods (which use conduction or convection), microwaves transfer heat directly via radiation to the food’s water molecules.
• Resonance ensures efficient energy transfer, as the microwave frequency matches the natural frequency of water molecules for optimal energy absorption.
• The principle of forced oscillations and resonance allows microwaves to heat food more efficiently than traditional methods.

Question 22

force field

Answer 22

• any region of space in which a specific type of 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

• Gravitational field lines (or vectors) show the direction of the gravitational force on a mass and always point toward the center of mass due to the attractive nature of gravity.
• The field around a point mass is radial, with lines converging toward the center.
• Field lines also represent the direction of acceleration for a mass placed in the field.

Question 25

Radial Vs Uniform field lines

Answer 25

• Radial fields are non-uniform, with field strength (g) varying with distance from the center; represented by radially inward lines.
• Uniform fields have equally spaced parallel lines, with field strength (g) remaining constant throughout.
• 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) and inversely proportional to the square of their separation (r).
• The negative sign indicates that the gravitational force is attractive.
• The distance r is measured between the centers of the two masses. Add the planet’s radius to the height above the surface when calculating r.

Question 27

Gravitational field strength in radial

Answer 27

• Gravitational field strength (g) in a radial field decreases with distance (r) according to the equation:
  g = -GM/r², where G is the gravitational constant, M is the mass, and r is the distance from the center of the mass.
• The negative sign shows that the field is attractive, and field lines point toward the mass causing the field.
• g decreases as r increases 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 is the work done per unit mass in bringing a mass from infinity to a defined point.
• Gravitational potential energy (G.P.E.) is the energy an object has due to its position in a gravitational field.
• Gravitational potential is always negative because work is required to move mass away from the gravitational source (attractive)
• Infinity is the reference point where the gravitational potential is defined as zero.
• The potential becomes less negative as mass is moved away from the source.

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 is the minimum speed required for an object to escape a gravitational field without further energy input.
• It depends on the mass and radius of the object creating the gravitational field and is the same for all objects in the same field.
• The escape velocity can be calculated using the equation v = √(2GM/r), where v is the escape velocity, G is the gravitational constant, M is the mass of the object creating the field, and r is the distance from the object’s center.
• An object reaches escape velocity when all its kinetic energy has been transferred to gravitational potential energy.

Question 31

Centripetal Force on a Planet

Answer 31

• The gravitational force on a satellite is centripetal, meaning it is always perpendicular to the direction of travel.
• Equating the gravitational force to the centripetal force gives:
  F = Fcentripetal.
• This simplifies to the equation for the orbital speed v of a satellite in a circular orbit:
  v = √(GM/r), where G is the gravitational constant, M is the mass of the central object, and r is the distance from the center of the object to the satellite.
• The mass of the satellite m cancels out from both sides of the equation, showing that orbital speed does not depend on 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

• A force-distance graph for gravitational fields would be a curve since force (F) is inversely proportional to r².
• The area under the graph represents work done or energy transferred.
• When a mass m moves further away from a mass M, its gravitational potential energy increases, requiring work to be done on the mass.
• 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 is a special orbit where 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 from west to east, with an orbital period of 24 hours, matching Earth’s rotational period.
• Used for telecommunication transmissions and television broadcasts, where the satellite amplifies and sends signals back to Earth.
• Receiver dishes on the Earth’s surface 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. Planet orbits can be circular or highly elliptical.
• Kepler’s Second Law: A line segment joining a planet and the Sun sweeps out equal areas in equal time intervals, meaning planets move faster when closer to the Sun and slower when farther away.
• Kepler’s Third Law: The square of the orbital time period (T) is directly proportional to the cube of the orbital radius (r), expressed as T² ∝ r³, with a constant (k) for 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) = S - mg (difference between upward force S and weight mg).
- Top of the Loop:
- Centripetal Force (F) = S + mg (sum of upward force S and weight mg).
- Side of the Loop:
- Centripetal Force (F) = S (equal to the upward force).

Circular Motion on a Banked Track
- Forces on a Banked Track:
- Centripetal force results from the combination of:
- Weight (mg): Acts vertically downward.
- Normal Force (N): Acts perpendicular to the surface.
- Components of Normal Force (N):
- Vertical Component: Balances weight
- Horizontal Component: Provides centripetal force
- Angle of Banking (θ):
- The angle determines the ratio of the normal force’s components, reducing reliance on friction.

Question 37

SHM in a pendulum

Answer 37

If given the height difference for pendulum, this is NOT amplitude!
The only thing you cao do with this is mgh = KE
WA=V max