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55-485: The newton-metre or newton-meter (also non-hyphenated, newton metre or newton meter ; symbol N⋅m or N m ) is the unit of torque (also called moment ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applied perpendicularly to the end of a moment arm that is one metre long. The unit is also used less commonly as

110-566: A ( t , r 0 ) − ω ( t ) × v ( t , r 0 ) = ψ c ( t ) + α ( t ) × A ( t ) r 0 {\displaystyle {\boldsymbol {\psi }}(t,\mathbf {r} _{0})=\mathbf {a} (t,\mathbf {r} _{0})-{\boldsymbol {\omega }}(t)\times \mathbf {v} (t,\mathbf {r} _{0})={\boldsymbol {\psi }}_{c}(t)+{\boldsymbol {\alpha }}(t)\times A(t)\mathbf {r} _{0}} where In 2D,

165-534: A force is allowed to act through a distance, it is doing mechanical work . Similarly, if torque is allowed to act through an angular displacement, it is doing work. Mathematically, for rotation about a fixed axis through the center of mass , the work W can be expressed as W = ∫ θ 1 θ 2 τ   d θ , {\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}\tau \ \mathrm {d} \theta ,} where τ

220-467: A given quantity expressed in newton-metres is a torque or a quantity of energy. "Even though torque has the same dimension as energy (SI unit joule), the joule is never used for expressing torque". Newton-metres and joules are dimensionally equivalent in the sense that they have the same expression in SI base units , but are distinguished in terms of applicable kind of quantity , to avoid misunderstandings when

275-403: A rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the same velocity . However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be

330-418: A rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non- collinear particles. This makes it possible to reconstruct the position of all

385-762: A single point particle is: L = r × p {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} } where p is the particle's linear momentum and r is the position vector from the origin. The time-derivative of this is: d L d t = r × d p d t + d r d t × p . {\displaystyle {\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\times \mathbf {p} .} This result can easily be proven by splitting

440-459: A time interval in which the orbit is completed once, is an integer times 360°. This integer is the winding number with respect to the origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon . Any point that is rigidly connected to the body can be used as reference point (origin of coordinate system L ) to describe the linear motion of the body (the linear position, velocity and acceleration vectors depend on

495-457: A torque is mistaken for an energy or vice versa. Similar examples of dimensionally equivalent units include Pa versus J/m, Bq versus Hz , and ohm versus ohm per square . Torque The term torque (from Latin torquēre , 'to twist') is said to have been suggested by James Thomson and appeared in print in April, 1884. Usage is attested the same year by Silvanus P. Thompson in

550-492: A twist applied to turn a shaft is better than the more complex notion of applying a linear force (or a pair of forces) with a certain leverage. Today, torque is referred to using different vocabulary depending on geographical location and field of study. This article follows the definition used in US physics in its usage of the word torque . In the UK and in US mechanical engineering , torque

605-411: A unit of work , or energy , in which case it is equivalent to the more common and standard SI unit of energy, the joule . In this usage the metre term represents the distance travelled or displacement in the direction of the force, and not the perpendicular distance from a fulcrum as it does when used to express torque. This usage is generally discouraged, since it can lead to confusion as to whether

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660-648: Is a general proof for point particles, but it can be generalized to a system of point particles by applying the above proof to each of the point particles and then summing over all the point particles. Similarly, the proof can be generalized to a continuous mass by applying the above proof to each point within the mass, and then integrating over the entire mass. In physics , rotatum is the derivative of torque with respect to time P = d τ d t , {\displaystyle \mathbf {P} ={\frac {\mathrm {d} {\boldsymbol {\tau }}}{\mathrm {d} t}},} where τ

715-608: Is any arbitrary point fixed in reference frame N, and the N to the left of the d/d t operator indicates that the derivative is taken in reference frame N. The result is independent of the selection of O so long as O is fixed in N. The acceleration of point P in reference frame N is defined as the time derivative in N of its velocity: For two points P and Q that are fixed on a rigid body B, where B has an angular velocity N ω B {\displaystyle \scriptstyle {^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }}} in

770-407: Is defined as the product of the magnitude of the perpendicular component of the force and the distance of the line of action of a force from the point around which it is being determined. In three dimensions, the torque is a pseudovector ; for point particles , it is given by the cross product of the displacement vector and the force vector. The direction of the torque can be determined by using

825-466: Is guaranteed by the Euler's rotation theorem ). All points on a rigid body experience the same angular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation . The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity

880-528: Is moving in frame N is given by where Q is the point fixed in B that instantaneously coincident with R at the instant of interest. This equation is often combined with Acceleration of two points fixed on a rigid body . If C is the origin of a local coordinate system L , attached to the body, the spatial or twist acceleration of a rigid body is defined as the spatial acceleration of C (as opposed to material acceleration above): ψ ( t , r 0 ) =

935-439: Is normal to the plane of symmetry and directed rightward, and b 3 is given by the cross product b 3 = b 1 × b 2 {\displaystyle b_{3}=b_{1}\times b_{2}} . In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation , respectively. Indeed,

990-495: Is not the time rate of change of orientation, because there is no such concept as an orientation vector that can be differentiated to obtain the angular velocity. The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D: In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable. For any set of three points P, Q, and R,

1045-462: Is referred to as moment of force , usually shortened to moment . This terminology can be traced back to at least 1811 in Siméon Denis Poisson 's Traité de mécanique . An English translation of Poisson's work appears in 1842. A force applied perpendicularly to a lever multiplied by its distance from the lever's fulcrum (the length of the lever arm ) is its torque. Therefore, torque

1100-425: Is the moment of inertia of the body and ω is its angular speed . Power is the work per unit time , given by P = τ ⋅ ω , {\displaystyle P={\boldsymbol {\tau }}\cdot {\boldsymbol {\omega }},} where P is power, τ is torque, ω is the angular velocity , and ⋅ {\displaystyle \cdot } represents

1155-405: Is the newton-metre (N⋅m). For more on the units of torque, see § Units . The net torque on a body determines the rate of change of the body's angular momentum , τ = d L d t {\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}} where L is the angular momentum vector and t

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1210-1748: Is time. For the motion of a point particle, L = I ω , {\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},} where I = m r 2 {\textstyle I=mr^{2}} is the moment of inertia and ω is the orbital angular velocity pseudovector. It follows that τ n e t = I 1 ω 1 ˙ e 1 ^ + I 2 ω 2 ˙ e 2 ^ + I 3 ω 3 ˙ e 3 ^ + I 1 ω 1 d e 1 ^ d t + I 2 ω 2 d e 2 ^ d t + I 3 ω 3 d e 3 ^ d t = I ω ˙ + ω × ( I ω ) {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=I_{1}{\dot {\omega _{1}}}{\hat {\boldsymbol {e_{1}}}}+I_{2}{\dot {\omega _{2}}}{\hat {\boldsymbol {e_{2}}}}+I_{3}{\dot {\omega _{3}}}{\hat {\boldsymbol {e_{3}}}}+I_{1}\omega _{1}{\frac {d{\hat {\boldsymbol {e_{1}}}}}{dt}}+I_{2}\omega _{2}{\frac {d{\hat {\boldsymbol {e_{2}}}}}{dt}}+I_{3}\omega _{3}{\frac {d{\hat {\boldsymbol {e_{3}}}}}{dt}}=I{\boldsymbol {\dot {\omega }}}+{\boldsymbol {\omega }}\times (I{\boldsymbol {\omega }})} using

1265-451: Is torque, and θ 1 and θ 2 represent (respectively) the initial and final angular positions of the body. It follows from the work–energy principle that W also represents the change in the rotational kinetic energy E r of the body, given by E r = 1 2 I ω 2 , {\displaystyle E_{\mathrm {r} }={\tfrac {1}{2}}I\omega ^{2},} where I

1320-814: Is torque. This word is derived from the Latin word rotātus meaning 'to rotate', but the term rotatum is not universally recognized but is commonly used. There is not a universally accepted lexicon to indicate the successive derivatives of rotatum, even if sometimes various proposals have been made. Using the cross product definition of torque, an alternative expression for rotatum is: P = r × d F d t + d r d t × F . {\displaystyle \mathbf {P} =\mathbf {r} \times {\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\times \mathbf {F} .} Because

1375-513: Is usually considered as a continuous distribution of mass . In the study of special relativity , a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light . In quantum mechanics , a rigid body is usually thought of as a collection of point masses . For instance, molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors ). The position of

1430-506: Is valid for any type of trajectory. In some simple cases like a rotating disc, where only the moment of inertia on rotating axis is, the rotational Newton's second law can be τ = I α {\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}} where α = ω ˙ {\displaystyle {\boldsymbol {\alpha }}={\dot {\boldsymbol {\omega }}}} . The definition of angular momentum for

1485-624: Is zero because velocity and momentum are parallel, so the second term vanishes. Therefore, torque on a particle is equal to the first derivative of its angular momentum with respect to time. If multiple forces are applied, according Newton's second law it follows that d L d t = r × F n e t = τ n e t . {\displaystyle {\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times \mathbf {F} _{\mathrm {net} }={\boldsymbol {\tau }}_{\mathrm {net} }.} This

1540-484: The geometrical theorem of the same name) states that the resultant torques due to several forces applied to about a point is equal to the sum of the contributing torques: Rigid body In physics , a rigid body , also known as a rigid object , is a solid body in which deformation is zero or negligible. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body

1595-460: The right hand grip rule : if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque. It follows that the torque vector is perpendicular to both the position and force vectors and defines the plane in which the two vectors lie. The resulting torque vector direction is determined by the right-hand rule. Therefore any force directed parallel to

1650-414: The scalar product . Algebraically, the equation may be rearranged to compute torque for a given angular speed and power output. The power injected by the torque depends only on the instantaneous angular speed – not on whether the angular speed increases, decreases, or remains constant while the torque is being applied (this is equivalent to the linear case where the power injected by a force depends only on

1705-721: The above expression for work, , gives W = ∫ s 1 s 2 F ⋅ d θ × r {\displaystyle W=\int _{s_{1}}^{s_{2}}\mathbf {F} \cdot \mathrm {d} {\boldsymbol {\theta }}\times \mathbf {r} } The expression inside the integral is a scalar triple product F ⋅ d θ × r = r × F ⋅ d θ {\displaystyle \mathbf {F} \cdot \mathrm {d} {\boldsymbol {\theta }}\times \mathbf {r} =\mathbf {r} \times \mathbf {F} \cdot \mathrm {d} {\boldsymbol {\theta }}} , but as per

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1760-418: The angular velocity is a scalar, and matrix A(t) simply represents a rotation in the xy -plane by an angle which is the integral of the angular velocity over time. Vehicles , walking people, etc., usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over

1815-407: The body (i.e. rotates together with the body), relative to another basis set (or coordinate system), from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal unit vectors b 1 , b 2 , b 3 , such that b 1 is parallel to the chord line of the wing and directed forward, b 2

1870-429: The choice). However, depending on the application, a convenient choice may be: When the center of mass is used as reference point: Two rigid bodies are said to be different (not copies) if there is no proper rotation from one to the other. A rigid body is called chiral if its mirror image is different in that sense, i.e., if it has either no symmetry or its symmetry group contains only proper rotations. In

1925-439: The definition of torque, and since the parameter of integration has been changed from linear displacement to angular displacement, the equation becomes W = ∫ θ 1 θ 2 τ ⋅ d θ {\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}{\boldsymbol {\tau }}\cdot \mathrm {d} {\boldsymbol {\theta }}} If

1980-411: The derivative of a vector is d e i ^ d t = ω × e i ^ {\displaystyle {d{\boldsymbol {\hat {e_{i}}}} \over dt}={\boldsymbol {\omega }}\times {\boldsymbol {\hat {e_{i}}}}} This equation is the rotational analogue of Newton's second law for point particles, and

2035-462: The equation for the Velocity of two points fixed on a rigid body in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as where N α B {\displaystyle \scriptstyle {{}^{\mathrm {N} }\!{\boldsymbol {\alpha }}^{\mathrm {B} }}} is the angular acceleration of B in

2090-544: The first edition of Dynamo-Electric Machinery . Thompson motivates the term as follows: Just as the Newtonian definition of force is that which produces or tends to produce motion (along a line), so torque may be defined as that which produces or tends to produce torsion (around an axis). It is better to use a term which treats this action as a single definite entity than to use terms like " couple " and " moment ", which suggest more complex ideas. The single notion of

2145-568: The infinitesimal linear displacement d s {\displaystyle \mathrm {d} \mathbf {s} } is related to a corresponding angular displacement d θ {\displaystyle \mathrm {d} {\boldsymbol {\theta }}} and the radius vector r {\displaystyle \mathbf {r} } as d s = d θ × r {\displaystyle \mathrm {d} \mathbf {s} =\mathrm {d} {\boldsymbol {\theta }}\times \mathbf {r} } Substitution in

2200-586: The instantaneous speed – not on the resulting acceleration, if any). The work done by a variable force acting over a finite linear displacement s {\displaystyle s} is given by integrating the force with respect to an elemental linear displacement d s {\displaystyle \mathrm {d} \mathbf {s} } W = ∫ s 1 s 2 F ⋅ d s {\displaystyle W=\int _{s_{1}}^{s_{2}}\mathbf {F} \cdot \mathrm {d} \mathbf {s} } However,

2255-446: The motion of a rigid body, such as linear and angular velocity , acceleration , momentum , impulse , and kinetic energy . The linear position can be represented by a vector with its tail at an arbitrary reference point in space (the origin of a chosen coordinate system ) and its tip at an arbitrary point of interest on the rigid body, typically coinciding with its center of mass or centroid . This reference point may define

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2310-498: The opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated version. The latter applies for S 2n , of which the case n = 1 is inversion symmetry. For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on

2365-403: The origin of a coordinate system fixed to the body. There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles , a quaternion , or a direction cosine matrix (also referred to as a rotation matrix ). All these methods actually define the orientation of a basis set (or coordinate system ) which has a fixed orientation relative to

2420-436: The other particles, provided that their time-invariant position relative to the three selected particles is known. However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by: Thus, the position of a rigid body has two components: linear and angular , respectively. The same is true for other kinematic and kinetic quantities describing

2475-426: The other side an image such that what shines through is the image at the top side, upside down. We can distinguish two cases: A sheet with a through and through image is achiral. We can distinguish again two cases: The configuration space of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlying manifold of the rotation group SO(3) . The configuration space of

2530-683: The particle's position vector does not produce a torque. The magnitude of torque applied to a rigid body depends on three quantities: the force applied, the lever arm vector connecting the point about which the torque is being measured to the point of force application, and the angle between the force and lever arm vectors. In symbols: τ = r × F ⟹ τ = r F ⊥ = r F sin ⁡ θ {\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} \implies \tau =rF_{\perp }=rF\sin \theta } where The SI unit for torque

2585-400: The point R is moving in the rigid body B while B moves in reference frame N, then the velocity of R in N is where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest. This relation is often combined with the relation for the Velocity of two points fixed on a rigid body . The acceleration in reference frame N of the point R moving in body B while B

2640-401: The position of a rigid body can be viewed as a hypothetic translation and rotation (roto-translation) of the body starting from a hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion). Velocity (also called linear velocity ) and angular velocity are measured with respect to a frame of reference . The linear velocity of

2695-414: The position vector from P to R is the sum of the position vector from P to Q and the position vector from Q to R: The norm of a position vector is the spatial distance. Here the coordinates of all three vectors must be expressed in coordinate frames with the same orientation. The velocity of point P in reference frame N is defined as the time derivative in N of the position vector from O to P: where O

2750-528: The rate of change of force is yank Y {\textstyle \mathbf {Y} } and the rate of change of position is velocity v {\textstyle \mathbf {v} } , the expression can be further simplified to: P = r × Y + v × F . {\displaystyle \mathbf {P} =\mathbf {r} \times \mathbf {Y} +\mathbf {v} \times \mathbf {F} .} The law of conservation of energy can also be used to understand torque. If

2805-489: The reference frame N, the velocity of Q in N can be expressed as a function of the velocity of P in N: where r P Q {\displaystyle \mathbf {r} ^{\mathrm {PQ} }} is the position vector from P to Q., with coordinates expressed in N (or a frame with the same orientation as N.) This relation can be derived from the temporal invariance of the norm distance between P and Q. By differentiating

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2860-487: The reference frame N. As mentioned above , all points on a rigid body B have the same angular velocity N ω B {\displaystyle {}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }} in a fixed reference frame N, and thus the same angular acceleration N α B . {\displaystyle {}^{\mathrm {N} }{\boldsymbol {\alpha }}^{\mathrm {B} }.} If

2915-401: The same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation . Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating (the existence of this instantaneous axis

2970-875: The torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., τ ⋅ d θ = | τ | | d θ | cos ⁡ 0 = τ d θ {\displaystyle {\boldsymbol {\tau }}\cdot \mathrm {d} {\boldsymbol {\theta }}=\left|{\boldsymbol {\tau }}\right|\left|\mathrm {d} {\boldsymbol {\theta }}\right|\cos 0=\tau \,\mathrm {d} \theta } giving W = ∫ θ 1 θ 2 τ d θ {\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}\tau \,\mathrm {d} \theta } The principle of moments, also known as Varignon's theorem (not to be confused with

3025-733: The vectors into components and applying the product rule . But because the rate of change of linear momentum is force F {\textstyle \mathbf {F} } and the rate of change of position is velocity v {\textstyle \mathbf {v} } , d L d t = r × F + v × p {\displaystyle {\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times \mathbf {F} +\mathbf {v} \times \mathbf {p} } The cross product of momentum p {\displaystyle \mathbf {p} } with its associated velocity v {\displaystyle \mathbf {v} }

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