6.Kinetic energy and work

Is momentarily stopped by the spring by what distance

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Unformatted text preview: riable force shown in the right figure. m ∆W j frictionless ∆ W j = F j , avg ∆ x W= ∑F xi xf x ∆x j , avg ∆x → 0 W= The Department of Physics, CUHK ∫ xf xi F ( x ) dx The Department of Physics, CUHK 17 18 3 Work done by a general variable force Work done by a general variable force Work-kinetic energy theorem with a variable force Consider now a particle that is acted on by a 3D force r ˆ F = Fx ˆ + F y ˆ + F z k i j W= Let the particle move through an incremental displacement ∵ r ˆ d r = dx ˆ + dyˆ + dz k i j rr dW = F ⋅ d r = Fx dx + F y dy + Fz dz ∴ W= ∫r i dW = xf ∫x zf yi i yf zi Fx dx + ∫ F y dy + ∫ Fz dz xf F ( x ) dx = xi ∫ xf xi madx dv dv dv dx dv dx = = v dt dt dx dt dx dv madx = m vdx = mvdv dx madx = m W= rf ∫ ∫ vf vi mvdv = 1 mv 2 − 1 mv i2 f 2 2 In conclusion W = K f − K i = ∆ K The Department of Physics, CUHK The Department of Physics, CUHK 19 20 §9. Power Sample problem 7-10 r F = (3 x 2 N ) ˆ + ( 4 N ) ˆ , with x in meters, acts on a i j Force particle, changing only the kinetic energy of the particle. How much a work is done on the particle as it moves from coordinates (2m, 3m) to (3m, 0m)? Does the speed of the particle increase, decrease, or remain the same? The time rate at which the work is done by a force is said to power be the power due to the force. If a force does an amount of work W in the time Δt, the average power due to the force during that time is W ∆t Pavg = instantaneous The instantaneous power is the instantaneous time rate of doing work, which takes the form, P= The Department of Physics, CUHK dW dt The Department of Physics, CUHK 21 Power 22 Power SI unit of the power: watt(W) We can also express the rate at which a force does work on a particle in terms of that force and the particle’s velocity. 1 W= 1 J/s other unit: horsepower (hp) 1 hp= 746 W The work can be expressed as power multiplied by time r F r v dW F cos θ dx P= = dt dt x P = Fv cos θ or 1 kilowatt-hour = 1kW.h = 3.6×106 J The Department of Physics, CUHK rr P = F ⋅v Instantaneous power The Department of Physics, CUHK 23 24 4 §10. Review & summary Kinetic energy Review & summary Work and kinetic energy 1 2 K = mv 2 Work done by a spring force 1 2 W is energy transferred to or from an object via a force acting on the object. Work done by a constant force rr W = Fd cos θ = F ⋅ d 1 2 W s = kx − kx ∆K = K f − K i = W Power 2 f Pavg = W ∆t average If xi=0, Work done by Fg Work 2 i W s = − 1 kx 2 2 W g = mgd cos θ P= Work done by a variable force Spring force Fx = − kx xf yf zf xi yi zi W = ∫ Fx dx + ∫ Fy dy + ∫ Fz dz Hooke ’s law The Department of Physics, CUHK dW dt instantaneous Power in terms of force and velocity velocity rr P = Fv cos θ = F ⋅ v The Department of Physics, CUHK 25 26 5...
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