Torque and Angular Acceleration

5 Key excerpts on "Torque and Angular Acceleration"

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  eBook - ePub

  Mechatronic Systems, Sensors, and Actuators

  Fundamentals and Modeling

  • Robert H. Bishop(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  1 ].

  20.4.1  Fundamental Concepts 20.4.1.1  Angular Displacement, Velocity, and Acceleration

  The concept of rotational motion is readily formalized: all points within a rotating rigid body move in parallel or coincident planes while remaining at fixed distances from a line called the axis. In a perfectly rigid body, all points also remain at fixed distances from each other. Rotation is perceived as a change in the angular position of a reference point on the body, i.e., as its angular displacement, Δθ, over some time interval, Δt. The motion of that point, and therefore of the whole body, is characterized by its clockwise (CW) or counterclockwise (CCW) direction and by its angular velocity, ω = Δθ/Δt. If during a time interval Δt, the velocity changes by Δω, the body is undergoing an angular acceleration, α = Δω/Δt. With angles measured in radians, and time in seconds, units of ω become radians per second (rad s−1 ) and of α, radians per second per second (rad s−2 ). Angular velocity is often referred to as rotational speed and measured in numbers of complete revolutions per minute (rpm) or per second (rps).

  20.4.1.2  Force, Torque, and Equilibrium

  Rotational motion, as with motion in general, is controlled by forces in accordance with Newton’s laws. Because a force directly affects only that component of motion in its line of action, forces or components of forces acting in any plane that includes the axis produce no tendency for rotation about that axis. Rotation can be initiated, altered in velocity, or terminated only by a tangential force Ft acting at a finite radial distance l from the axis. The effectiveness of such forces increases with both Ft and l; hence, their product, called a moment, is the activating quantity for rotational motion. A moment about the rotational axis constitutes a torque. Figure 20.45 (a) shows a force F acting at an angle β to the tangent at a point P, distant l (the moment arm) from the axis. The torque T is found from the tangential component of F

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  BIOS Instant Notes in Sport and Exercise Biomechanics

  • Paul Grimshaw, Neil Fowler, Adrian Lees, Adrian Burden(Authors)
  • 2007(Publication Date)
  • Routledge

    (Publisher)

  Section C – Kinetics of angular motion

  C1	TORQUE AND THE MOMENT OF FORCE

  Key Notes

  Torque	A torque is a twisting or turning moment that is calculated by multiplying the force applied by the perpendicular distance (from the axis of rotation) at which the force acts (the moment arm). Torques cause angular accelerations that result in rotational movement of limbs/segments.
  Clockwise and anti-clockwise rotation	Clockwise rotation is the rotary movement of a limb/lever/segment in a clockwise direction (−ve). Clockwise is referring, in this case, to the hands of a clock or watch. Anti-clockwise rotation is rotary movement in the opposite direction (+ve).
  Force couple	A force couple is a pair of equal and opposite parallel forces acting on a system.
  Equilibrium	This is a situation in which all the forces and moments acting are balanced, and which results in no rotational acceleration (i.e., a constant velocity situation).
  Second condition of equilibrium	This states that the sum of all the torques acting on an object is zero and the object does not change its rotational velocity. Re-written, this condition can be expressed as the sum of the anti-clockwise and clockwise moments acting on a system is equal to zero (∑ACWM + ∑CWM = 0).
  Application	Swimmers are now utilizing a pronounced bent elbow underwater pull pattern during the freestyle arm action. This recent technique change allows the swimmer to acquire more propulsive force and yet prevent excessive torques being applied to the shoulder joint (which were previously caused by a long arm pull underwater pattern). Large torques are needed at the hip joint (hip extensor and flexor muscles) to create the acceleration of the limbs needed to kick a soccer ball.

  Torque

  A torque is defined as a twisting or turning moment. The term moment is the force acting at a distance from an axis of rotation. Torque can therefore be calculated by multiplying the force applied by the perpendicular distance at which the force acts from the axis of rotation. Often the term torque is referred to as the moment of force. The moment of force is the tendency of a force to cause rotation about an axis. Torque is a vector quantity and as such it is expressed with both magnitude and direction. Within human movement or exercise science torques cause angular acceleration that result in the rotational movements of the limbs and segments

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  Engineering Science

  For Foundation Degree and Higher National

  • Mike Tooley, Lloyd Dingle(Authors)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  We considered angular motion in section 8.3. We now extend the notion of angular motion to include the accelerations and forces that act on bodies that move in a circular path, for example, motor vehicles that negotiate bends, aircraft that bank and turn or governors that have rotating bob weights to control the rate of opening of valves and the like. We start by considering the acceleration and forces that occur as a result of circular motion.

  8.8.1 Centripetal acceleration and force

  You will be aware of the relationship between linear and angular velocity through use of the transformation equations, where we know that s = rθ , v = rω and a = rα . We can use these formulae to form relationships for the accelerations and forces that act on a body during circular motion. Consider Figure 8.22 , where a point mass is subject to a rotational velocity.

  Figure 8.22 A point mass subject to a rotational velocity

  From the figure it can be seen that the direction of motion of the mass must be continually changing, in order to produce the circular motion. Therefore, it is being subjected to an acceleration that is acting towards the centre of rotation ; this is known as centripetal acceleration .

  Now, from the transformation equations we know that the tangential velocity v = r ω and that the acceleration of the mass is equal to its change in velocity divided by the time taken for the mass to move through some angular distance θ. Thus the change in the tangential velocity of the mass divided by the time, that is its acceleration , is given as and since the angular velocity is equal to the angle turned through in radian divided by time, Then, from above, the centripetal acceleration of the mass = ω2 r. Also, we know that v = rω or so that centripetal therefore we may write that:

  Now when this acceleration acts on a mass, as in this case, it produces a force known as centripetal force, thus: centripetal force (

  Fc

  ) = mass × centripetal acceleration . Then:

  Now, remembering Newton’s third law, that to every action there is an equal and opposite reaction, the centripetal force acting on the mass pushing it towards the centre of rotation must be balanced by the radial force created by the mass maintaining the orbit. This force opposing the centripetal force is known as the centrifugal force

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  Biomechanics For Dummies

  • Steve McCaw(Author)
  • 2014(Publication Date)
  • For Dummies

    (Publisher)

  V .

  A New Angle on Newton: Angular Versions of Newton's Laws

  Newton's laws of motion (see Chapter 6 ) explain that an unbalanced force causes an acceleration of a body, meaning the magnitude of the acceleration is proportional to the magnitude and direction of the unbalanced force and inversely proportional to the body's mass, and that forces equal in magnitude and opposite in direction are applied to the two bodies interacting to produce the force. In most human motion, the mass of the body stays constant, and only the motion of the body can change.

  Torque (see Chapter 8 ) is the turning effect of a force. If an applied force does not pass through a specified axis of rotation, the force has a moment arm around the axis and creates a torque on the body. When more than one torque acts around an axis, calculating the net torque at the axis (ΣT axis ) shows if an unbalanced torque (ΣT axis ≠ 0; see Chapter 8 for more on calculating net torque) is present around the axis. Newton's laws of motion explain the effect of an unbalanced torque on the angular motion of a body.

  The two main points to remember in looking at Newton's laws for angular motion are that the moment of inertia — the resistance to changing angular motion — is not necessarily constant for a body, and a force must have a moment arm to produce a turning effect on a body. In the following sections, I explain Newton's laws of angular motion.

  Maintaining angular momentum: Newton's first law

  Newton's first law, also known as the law of conservation of angular momentum, specifies that an unbalanced external torque causes a change in the angular momentum of a body. Angular momentum is the product of the moment of inertia (I ) and the angular velocity (ω), or H = I

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  Higher Engineering Science

  • William Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  19 A 200 mm diameter drive pulley rotates at 5 rev/s. It drives, via a belt drive, another pulley. What diameter will this need to be if it is required to rotate a shaft at 2.5 rev/s?

  4.4.4 Combined linear and angular motion

  Consider objects which have both a linear motion and angular motion, e.g. a rolling wheel. For a wheel of radius r which is rolling, without slip, along a straight path (Figure 4.24 ), when the wheel rotates and rolls its centre moves from C to C′ then a point on its rim moves from O to O′. The distance CC′ equals OO′ But OO′ = rθ. Thus:

  Figure 4.24 Rolling wheel

  horizontal distance moved by the wheel x = rπ

  [31]

  If this movement occurs in a time t, then differentiating equation [31 ]:

  and thus:

  horizontal velocity vx = rω

  [32]

  where ω is the angular velocity of the wheel. Differentiating equation [32 ] gives:

  and thus:

  horizontal acceleration = ra

  [33]

  where a is the angular acceleration.

  4.5 Force and linear motion

  In considering the effects of force on the motion of a body we use Newton’s laws.

  Newton’s laws can be expressed as:

  First law

  A body continues in its state of rest or uniform motion in a straight line unless acted on by a force.

  Second law

  The rate of change of momentum of a body is proportional to the applied force and takes place in the direction of the force.

  Third law

  When a body A exerts a force on a body B, B exerts an equal and opposite force on A (this is often expressed as: to every action there is an opposite and equal reaction).

  Thus the first law indicates that if we have an object moving with a constant velocity there can be no resultant force acting on it. If there is a resultant force, then the second law indicates that there will be a change in momentum with:

  [34]

  For a mass m which does not change with time:

  dv/dt is the acceleration. The units are chosen so that the constant of proportionality is 1 and thus the second law can be expressed as:

  F = ma

  [35]

  If a body of mass m is allowed to freely fall then it would fall with the acceleration due to gravity g. The force acting on the freely falling object is thus mg and this is termed its weight

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