• Consider two forces F1 and F2 having equal magnitude and opposite direction acting on a stick placed on a horizontal table as shown below in the figure

    Two forces acting on stick are equal in magnitude but acting in opposite direction tends to produce the torque

  • Here note that line of action of forces F1 and F2 is not same .So they tend to rotate the stick in clockwise direction
  • This tendency of the force to rotate an object about some axis is called torque
  • Torque is the rotational counterpart of force. torque tends to rotate an body in the same way as force tends to change the state of motion of the body
  • Figure below shows a rigid body pivoted at point O so that point O is fixed in space and the body is free to rotate

    force action of the rigid body pivoted at any point produces a torque

  • Let P be the point of application of force. This force acting at point P makes an angle θ with the radius vector r from point O to P
  • This force F can be resolved into two components
    as they are perpendicular and parallel to r
  • Parallel component of force does not produce rotational motion of body around point O as it passes through O
  • Effect of perpendicular components producing rotation of rigid body through point O depends on magnitude of the perpendicular force and on its distance r from O
  • Mathematically ,torque about point O is defined as product of perpendicular component of force and r i.e.
    τ=Fr=Fsinθr=F(rinθ)=Fd              ---(18)
    where d is the perpendicular distance from the pivot point ) to the line of action of force F
  • Quantity d=rinθ is called moment arm or liner arm of force F .If d=0 the there would be no rotation
  • Torque can either be anticlockwise or clockwise depending on the sense of rotation it tends to produce
  • Unit of torque is Nm
  • Consider the figure given below where a rigid body pivoted at point O is acted upon by the two force F1 and F2
  • d1 is the moment arm of force F1 and d2 is the moment arm of force F2

    Figure shows the different forces trying to rotate the body in different direction

  • Force F2 has the tendency to rotate rigid body in clockwise direction and F1 has the to rotate it in anti clockwise direction
  • Here we adopt a convention that anticlockwise moments are positive and clockwise moment are negative
  • hence moment τ1 of force F1 about the axis through O is
    And that of force F2 would be
  • Hence net torque about O is
    τtotal= τ1+ τ2
  • Rotation of the body can be prevented if
    or τ1=-τ2
  • We earlier studied that when a body is in equilibrium under the action of several coplanar forces ,the vector sum of these forces must be zero i.e.
    ΣFx=0 and ΣFy=0
  • we know state our second condition for static equilibrium of rigid bodies that is
    " For static equilibrium of rigid body net torque in clockwise direction must be equal to net torque in anticlockwise direction w.r.t some specified axis i.e.
  • Thus for static equilibrium of an rigid body
    i) The resultant external force must be zero
    ii) The resultant external torque about any point or axis of rotation must be zero i.e.

(8) work and power in rotational motion

  • We know that when we apply force on any object in direction of the displacement of the object ,work is said to be done
  • Similarly force applied to the rotational body does work on it and this work done can be expressed in terms of moment of force (torque) and angular displacement θ
  • Consider the figure given below where a force F acts on the wheel of radius R pivoted at point O .so that it can rotate through point O

    Figure shows the force action on wheel pivoted at point O

  • This force F rotates the wheel through an angle dθ and dθ is small enough so that we can regard force to be constant during corresponding time interval dt
  • Workdone by this force is
    but ds=Rdθ
  • Now FR is the torque Τ due to force we have
    dW=Τdθ ----(19)
  • if the torque is constant while angle changes from θ1 to θ2 then
    W=Τ(θ21)=ΤΔθ ---(20)
    Thus workdone by the constant torque equals the product of the torque and angular displacement
  • we know that rate of doing work is the power input of torque so
  • In vector notation

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