Contents 1 Units 2 Equations for power 3 Average power 4 Mechanical power 4.1 Mechanical advantage 5 Electrical power 6 Peak power and duty cycle 7 Radiant power 8 See also 9 References

Units[edit] The dimension of power is energy divided by time. The SI unit of power is the watt (W), which is equal to one joule per second. Other units of power include ergs per second (erg/s), horsepower (hp), metric horsepower (Pferdestärke (PS) or cheval vapeur (CV)), and foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 pounds by one foot in one second, and is equivalent to about 746 watts. Other units include dBm, a relative logarithmic measure with 1 milliwatt as reference; food calories per hour (often referred to as kilocalories per hour); BTU per hour (BTU/h); and tons of refrigeration (12,000 BTU/h).

Equations for power[edit] Power, as a function of time, is the rate at which work is done, so can be expressed by this equation: P = d W d t {\displaystyle P={\frac {dW}{dt}}} where P is power, W is work, and t is time. Because work is a force F applied over a distance r, W = F ⋅ r {\displaystyle W={\mathbf {F}}\cdot {\mathbf {r}}} for a constant force, power can be rewritten as: P = d W d t = d d t ( F ⋅ r ) = F ⋅ d r d t = F ⋅ v {\displaystyle P={\frac {dW}{dt}}={\frac {d}{dt}}\left({\mathbf {F}}\cdot {\mathbf {r}}\right)={\mathbf {F}}\cdot {\frac {d{\mathbf {r}}}{dt}}={\mathbf {F}}\cdot {\mathbf {v}}}

Average power[edit] As a simple example, burning one kilogram of coal releases much more energy than does detonating a kilogram of TNT,[3] but because the TNT reaction releases energy much more quickly, it delivers far more power than the coal. If ΔW is the amount of work performed during a period of time of duration Δt, the average power Pavg over that period is given by the formula P a v g = Δ W Δ t . {\displaystyle P_{\mathrm {avg} }={\frac {\Delta W}{\Delta t}}\,.} It is the average amount of work done or energy converted per unit of time. The average power is often simply called "power" when the context makes it clear. The instantaneous power is then the limiting value of the average power as the time interval Δt approaches zero. P = lim Δ t → 0 P a v g = lim Δ t → 0 Δ W Δ t = d W d t . {\displaystyle P=\lim _{\Delta t\rightarrow 0}P_{\mathrm {avg} }=\lim _{\Delta t\rightarrow 0}{\frac {\Delta W}{\Delta t}}={\frac {\mathrm {d} W}{\mathrm {d} t}}\,.} In the case of constant power P, the amount of work performed during a period of duration T is given by: W = P t . {\displaystyle W=Pt\,.} In the context of energy conversion, it is more customary to use the symbol E rather than W.

Mechanical power[edit] One metric horsepower is needed to lift 75 kilograms by 1 meter in 1 second. Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity. Mechanical power is also described as the time derivative of work. In mechanics, the work done by a force F on an object that travels along a curve C is given by the line integral: W C = ∫ C F ⋅ v d t = ∫ C F ⋅ d x , {\displaystyle W_{C}=\int _{C}{\mathbf {F}}\cdot {\mathbf {v}}\,\mathrm {d} t=\int _{C}{\mathbf {F}}\cdot \mathrm {d} {\mathbf {x}},} where x defines the path C and v is the velocity along this path. If the force F is derivable from a potential (conservative), then applying the gradient theorem (and remembering that force is the negative of the gradient of the potential energy) yields: W C = U ( B ) − U ( A ) , {\displaystyle W_{C}=U(B)-U(A),} where A and B are the beginning and end of the path along which the work was done. The power at any point along the curve C is the time derivative P ( t ) = d W d t = F ⋅ v = − d U d t . {\displaystyle P(t)={\frac {\mathrm {d} W}{\mathrm {d} t}}={\mathbf {F}}\cdot {\mathbf {v}}=-{\frac {\mathrm {d} U}{\mathrm {d} t}}.} In one dimension, this can be simplified to: P ( t ) = F ⋅ v . {\displaystyle P(t)=F\cdot v.} In rotational systems, power is the product of the torque τ and angular velocity ω, P ( t ) = τ ⋅ ω , {\displaystyle P(t)={\boldsymbol {\tau }}\cdot {\boldsymbol {\omega }},\,} where ω measured in radians per second. The ⋅ {\displaystyle \cdot } represents scalar product. In fluid power systems such as hydraulic actuators, power is given by P ( t ) = p Q , {\displaystyle P(t)=pQ,\!} where p is pressure in pascals, or N/m2 and Q is volumetric flow rate in m3/s in SI units. Mechanical advantage[edit] If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for the mechanical advantage of the system. Let the input power to a device be a force FA acting on a point that moves with velocity vA and the output power be a force FB acts on a point that moves with velocity vB. If there are no losses in the system, then P = F B v B = F A v A , {\displaystyle P=F_{B}v_{B}=F_{A}v_{A},\!} and the mechanical advantage of the system (output force per input force) is given by M A = F B F A = v A v B . {\displaystyle \mathrm {MA} ={\frac {F_{B}}{F_{A}}}={\frac {v_{A}}{v_{B}}}.} The similar relationship is obtained for rotating systems, where TA and ωA are the torque and angular velocity of the input and TB and ωB are the torque and angular velocity of the output. If there are no losses in the system, then P = T A ω A = T B ω B , {\displaystyle P=T_{A}\omega _{A}=T_{B}\omega _{B},\!} which yields the mechanical advantage M A = T B T A = ω A ω B . {\displaystyle \mathrm {MA} ={\frac {T_{B}}{T_{A}}}={\frac {\omega _{A}}{\omega _{B}}}.} These relations are important because they define the maximum performance of a device in terms of velocity ratios determined by its physical dimensions. See for example gear ratios.

Electrical power[edit] Ansel Adams photograph of electrical wires of the Boulder Dam Power Units, 1941–1942 Main article: Electric power The instantaneous electrical power P delivered to a component is given by P ( t ) = I ( t ) ⋅ V ( t ) {\displaystyle P(t)=I(t)\cdot V(t)\,} where P(t) is the instantaneous power, measured in watts (joules per second) V(t) is the potential difference (or voltage drop) across the component, measured in volts I(t) is the current through it, measured in amperes If the component is a resistor with time-invariant voltage to current ratio, then: P = I ⋅ V = I 2 ⋅ R = V 2 R {\displaystyle P=I\cdot V=I^{2}\cdot R={\frac {V^{2}}{R}}\,} where R = V I {\displaystyle R={\frac {V}{I}}\,} is the resistance, measured in ohms.

Peak power and duty cycle[edit] In a train of identical pulses, the instantaneous power is a periodic function of time. The ratio of the pulse duration to the period is equal to the ratio of the average power to the peak power. It is also called the duty cycle (see text for definitions). In the case of a periodic signal s ( t ) {\displaystyle s(t)} of period T {\displaystyle T} , like a train of identical pulses, the instantaneous power p ( t ) = | s ( t ) | 2 {\displaystyle p(t)=|s(t)|^{2}} is also a periodic function of period T {\displaystyle T} . The peak power is simply defined by: P 0 = max [ p ( t ) ] {\displaystyle P_{0}=\max[p(t)]} . The peak power is not always readily measurable, however, and the measurement of the average power P a v g {\displaystyle P_{\mathrm {avg} }} is more commonly performed by an instrument. If one defines the energy per pulse as: ϵ p u l s e = ∫ 0 T p ( t ) d t {\displaystyle \epsilon _{\mathrm {pulse} }=\int _{0}^{T}p(t)\mathrm {d} t\,} then the average power is: P a v g = 1 T ∫ 0 T p ( t ) d t = ϵ p u l s e T {\displaystyle P_{\mathrm {avg} }={\frac {1}{T}}\int _{0}^{T}p(t)\mathrm {d} t={\frac {\epsilon _{\mathrm {pulse} }}{T}}\,} . One may define the pulse length τ {\displaystyle \tau } such that P 0 τ = ϵ p u l s e {\displaystyle P_{0}\tau =\epsilon _{\mathrm {pulse} }} so that the ratios P a v g P 0 = τ T {\displaystyle {\frac {P_{\mathrm {avg} }}{P_{0}}}={\frac {\tau }{T}}\,} are equal. These ratios are called the duty cycle of the pulse train.

Radiant power[edit] Power is related to intensity at a distance r, the power emitted by a source can be written as:[citation needed] P ( r ) = I ( 4 ∗ p i ∗ r 2 ) {\displaystyle P(r)=I(4*pi*r^{2})}

See also[edit] Simple machines Motive power Orders of magnitude (power) Pulsed power Intensity — in the radiative sense, power per area Power gain — for linear, two-port networks. Power density Signal strength Sound power

References[edit] ^ Halliday and Resnick (1974). "6. Power". Fundamentals of Physics. CS1 maint: Uses authors parameter (link) ^ Chapter 13, § 3, pp 13-2,3 The Feynman Lectures on Physics Volume I, 1963 ^ Burning coal produces around 15-30 megajoules per kilogram, while detonating TNT produces about 4.7 megajoules per kilogram. For the coal value, see Fisher, Juliya (2003). "Energy Density of Coal". The Physics Factbook. Retrieved 30 May 2011.  For the TNT value, see the article TNT equivalent. Neither value includes the weight of oxygen from the air used during combustion. v t e Classical mechanics SI units Linear/translational quantities Angular/rotational quantities Dimensions 1 L L2 Dimensions 1 1 1 T time: t s absement: A m s T time: t s 1 distance: d, position: r, s, x, displacement m area: A m2 1 angle: θ, angular displacement: θ rad solid angle: Ω rad2, sr T−1 frequency: f s−1, Hz speed: v, velocity: v m s−1 kinematic viscosity: ν, specific angular momentum: h m2 s−1 T−1 frequency: f s−1, Hz angular speed: ω, angular velocity: ω rad s−1 T−2 acceleration: a m s−2 T−2 angular acceleration: α rad s−2 T−3 jerk: j m s−3 T−3 angular jerk: ζ rad s−3 M mass: m kg ML2 moment of inertia: I kg m2 MT−1 momentum: p, impulse: J kg m s−1, N s action: 𝒮, actergy: ℵ kg m2 s−1, J s ML2T−1 angular momentum: L, angular impulse: ΔL kg m2 s−1 action: 𝒮, actergy: ℵ kg m2 s−1, J s MT−2 force: F, weight: Fg kg m s−2, N energy: E, work: W kg m2 s−2, J ML2T−2 torque: τ, moment: M kg m2 s−2, N m energy: E, work: W kg m2 s−2, J MT−3 yank: Y kg m s−3, N s−1 power: P kg m2 s−3, W ML2T−3 rotatum: P kg m2 s−3, N m s−1 power: P kg m2 s−3, W Retrieved from "" Categories: Concepts in physicsPower (physics)Hidden categories: CS1 maint: Uses authors parameterUse dmy dates from July 2012All articles with unsourced statementsArticles with unsourced statements from August 2017

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