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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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