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Chapter 10

Chapter 10. Rotation. Rotation. Most motion we have discussed thus far refers to translation. Now we discuss the mechanics of ROTATION , describing motion in a circle . First, we must define the standard rotational properties. A RIGID BODY refers to one where all the parts rotate

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Chapter 10

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  1. Chapter 10 Rotation

  2. Rotation Most motion we have discussed thus far refers to translation. Now we discuss the mechanics of ROTATION, describing motion in a circle. First, we must define the standard rotational properties. A RIGID BODY refers to one where all the parts rotate about a given axis without changing its shape. (Note that in pure translation, each point moves the same linear distance during a particular time interval). A fixed axis, known as the AXIS OF ROTATION is defined by one that does not change position under rotation. Each point on the body moves in a circular path described by an angular displacement Dq. The origin of this circular path is centred at the axis of rotation. REGAN PHY34210

  3. axis of rotation reference line r q s Summary of Rotational Variables All rotational variables are defined relative to motion about a fixed axis of rotation. The ANGULAR POSITION, q, of a body is then the angle between a REFERENCE LINE, which is fixed in the body and perpendicular to the rotation axis relative to a fixed direction (e.g., the x-axis). If q is in radians, we know that q=s/r where s is the length of arc swept out by a radius r moving through an angle q. (Note counterclockwise represent increase in positive q). Radians are defined by s/r and are thus pure, dimensionless numbers without units. The circumference of a circle (i.e., a full arc) s=2pr, thus in radians, the angle swept out by a single, full revolution is 360o = 2pr/r=2p. Thus, 1 radian = 360 / 2p = 57.3o = 0.159 of a complete revolution. x REGAN PHY34210

  4. The angular displacement, Dq represents the change in the angular position due to rotational motion. In analogy with the translational motion variables, other angular motion variables can be defined in terms of the change (Dq),rate of change (w ) and rate of rate of change (a ) of the angular position. REGAN PHY34210

  5. Relating Linear and Angular Variables For the rotation of a rigid body, all of the particles in the body take the same time to complete one revolution, which means that they all have the same angular velocity,w, i.e., they sweep out the same measure of arc, dq in a given time. However, the distance travelled by each of the particles, s, differs dramatically depending on the distance,r, from the axis of rotation, with the particles with the furthest from the axis of rotation having the greatest speed, v. at and ar are the tangential and radial accelerations respectively. We can relate the rotational and linear variables using the following (NB.: RADIANS MUST BE USED FOR ANGULAR VARIABLES!) REGAN PHY34210

  6. Rotation with Constant Acceleration For translational motion we have seen that for the case of a constant acceleration, we can derive a series of equations of motion. By analogy, for CONSTANT ANGULAR ACCELERATION, there is a corresponding set of equations which can be derived by substituting the translational variable with its rotational analogue. ROTATIONAL TRANSLATIONAL REGAN PHY34210

  7. ref. line for q0=0 axis of rotation Example 1: A grindstone rotates at a constant angular acceleration of a=0.35rad/s2. At time t=0 it has an angular velocity of w0=-4.6rad/s and a reference line on its horizontal at the angular position, q0=0. (a) at what time after t=0 is the reference line at q=5 revs ? Note that while w0 is negative,a is positive. Thus the grindstone starts rotating in one direction, then slows with constant deceleration before changing direction and accelerating in the positive direction. At what time does the grindstone momentarily stop to reverse direction? REGAN PHY34210

  8. Kinetic Energy of Rotation REGAN PHY34210

  9. Calculating to the Rotational Moment of Inertia The Parallel-Axis Theorem To calculate I if the moment of inertia about a parallel axis passing through the body’s centre of mass is known, we can use I=Icom+Mh2, where, M= the total mass of the body, h is the perpendicular distance between the parallel centre of mass axis and the axis of rotation and Icom is the moment of inertia about the centre of mass axis. REGAN PHY34210

  10. a d-a Cl com H d rotation axis Example 2: The HCl molecule consists of a hydrogen atom (mass 1u) and a chlorine atom (mass 35u). The centres of the two atoms are separated by 127pm (=1.27x10-10m). What is the moment of inertia, I, about an axis perpendicular to the line joining the two atoms which passes through the centre of mass of the HCl molecule ? REGAN PHY34210

  11. F Ft Frad f r O Torque and Newton’s 2nd Law The ability of a force, F, to rotate an object depends not just on the magnitude of its tangential component, Ftbut also on how far the applied force is from the axis of rotation, r. The product of Ft r =Frsinf is called the TORQUE (latin for twist!)t . REGAN PHY34210

  12. Work and Rotational Kinetic Energy REGAN PHY34210

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