Gravitation, circular motion and oscillations
Constant with changing
includes direction as well as . A body moving at constant around a circle changes direction continuously and therefore accelerates. Uniform circular points toward the centre with magnitude. The required resultant inward is mv²/r. Centripetal is the role played by net inward , not a new added to tension, or gravity. Identify the actual provider: tension in a whirled string, gravity in an ideal orbit or in a level-road turn. If the net inward vanishes, the existing tangential does not automatically point radially outward.
Period and angular
Period T is one-revolution time and f revolutions per second, so. Circumference2πr divided by period gives. One revolution is2π radians; angular ω and. Circular can therefore beω²r. Points on a rigid rotating wheel share angular but a larger-radius point has greater linear . Angular is rate of change of angular in a fixed-axis simple model. Convert revolutions per minute to revolutions per second before using SI time. State whether a quantity describes angular change or linear path length.
Horizontal and vertical diagrams
On a horizontal frictionless table tension can all inward . In a vertical circle, gravity contributes differently at the top and bottom: at the top weight and tension can both point inward, while at the bottom tension points inward and weight outward. Tension is therefore not automatically constant. For a conical pendulum, vertical tension component balances weight and horizontal component supplies centripetal . The same tension is resolved into components; it is not three independent forces. An apparent centrifugal is used in a rotating reference frame, not added as an extra outward physical interaction when applying an inertial-frame diagram.
Universal gravitation and field strength
Two point masses, or suitable spherically symmetric bodies outside their surfaces, attract with, where r is centre-to-centre separation. Doubling r quarters at fixed masses. Gravitational field strength is per test mass in N/kg, equivalent to m/s². A test mass cancels when gravity equals ma, explaining mass-independent ideal free fall. Near Earth's surface g changes little over small heights, giving the constant-g approximation. Across the surface, , Earth's rotation and its nonspherical shape contribute to variation; at significant altitude the increased centre distance also reduces the gravitational field contribution.
Gravitational potential and conservative work
Near the surface small height changes useΔU. For large separations from an isolated spherical mass, with zero at infinity. Negative potential indicates a bound attractive system under that reference. Raising an object makes its potential less negative. Gravitational potential is per unit test mass, not field strength. Gravity is conservative: work between two positions depends on endpoints rather than path and the net work around a closed path is zero. is generally nonconservative and transfers mechanical to internal stores. Escape √(2GM/r) follows the ideal no- account and does not depend on launched mass.
Circular orbits and weightlessness
Equating gravity to centripetal gives and. Orbital radius includes planet radius plus altitude. The period is2π√(r³/GM), so a larger circular orbit has a longer period and smaller . Orbiting astronauts are still affected by gravity; they and the craft fall together, reducing ordinary support forces and producing apparent weightlessness. A geostationary orbit must have the appropriate Earth-rotation period, lie in the equatorial plane and follow Earth's rotation direction. A satellite with the same period but the wrong plane is not stationary over one point in the same sense.
Simple harmonic motion
SHM requires proportional to from and directed toward it. The restoring minus sign matters. is maximum at and zero at extreme displacements; magnitude is greatest at the extremes. A mass on an ideal spring has period. A simple small-angle pendulum has approximate period, with effective length measured from pivot to bob centre. A large swing violates the small-angle approximation. Amplitude is the maximum and differs from period and . Do not apply the spring formula to every periodic system merely because it moves back and forth.
, damping and resonance
Ideal oscillations exchange kinetic and potential while total mechanical remains constant. An ideal spring's total oscillation is½kA² for amplitude A. At kinetic is greatest; at extremes it is zero and potential greatest. Damping transfers away and reduces amplitude. A periodic driver can replace losses, and a driving near natural can create resonance with a large response depending on damping. The driver supplies that ; resonance does not create it. A bridge or instrument example should explain a driving mechanism and natural response rather than treating every vibration as necessarily resonant.
Testing a pendulum model
Measure several complete oscillations and divide total time by their number to reduce fractional reaction-time effects. Use small amplitude and a consistent crossing point, repeat measurements and vary length in a suitable range. The ideal relation gives a straight T²-versus-L graph with gradient4π²/g. A T-versus-L graph is curved. Measure to the bob's centre of mass, not its upper surface. Avoid pushing it into unwanted elliptical motion. A graph intercept or systematic discrepancy can reveal a length offset or other model limitation; it should not be dismissed as generic human error without explanation.
Self-check and proportional reasoning
A0.5 kg object at4 m/s in radius2 m needs inward force0.5 ×16/2 =4 N. If its string breaks, it initially follows a tangent subject to remaining forces. Doubling gravitational separation quarters attraction. Multiplying a small-angle pendulum length by four doubles its period at fixed g. These are different proportional laws: v²/r,1/r² and√L respectively. State which variables are held fixed in each comparison. In numerical work retain consistent SI units and interpret signs from an explicit frame. A negative gravitational potential is compatible with a positive escape- requirement because must be supplied to reach the zero-at-infinity reference.