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Topic | Discussion | Learning Aids
Line of action Lever arm | 9.1 THE ACTION OF FORCES AND TORQUES ON RIGID OBJECTS The line of action of a force is an extended line that is drawn colinear with the force. The lever arm is the distance between the line of action and the axis of rotation, measured on a line that is perpendicular to both. |
Torque | The torque of a force has a magnitude that is given by the magnitude F of the force times the lever arm . The torque is

| (9.1)

and is positive when the force tends to produce a counterclockwise rotation about the axis, and negative when the force tends to produce a clockwise rotation.
| 9.2 RIGID OBJECTS IN EQUILIBRIUM A rigid body is in equilibrium if it has zero translational acceleration and zero angular acceleration. In equilibrium, the net external force and the net external torque acting on the body are zero: |
Equilibrium of a rigid body |

| (4.9a and 4.9b)

| (9.2)

| 9.3 CENTER OF GRAVITY The center of gravity of a rigid object is the point where its entire weight can be considered to act when calculating the torque due to the weight. For a symmetrical body with uniformly distributed weight, the center of gravity is at the geometrical center of the body. When a number of objects whose weights are W1, W2, … are distributed along the x axis at locations x1, x2, …, the center of gravity xcg is located at |
Definition of center of gravity |

| (9.3)

The center of gravity is identical to the center of mass, provided the acceleration due to gravity does not vary over the physical extent of the objects.
Use Self-Assessment Test 9.1 to evaluate your understanding of Sections 9.1, 9.2 and 9.3.
| 9.4 NEWTON’S SECOND LAW FOR ROTATIONAL MOTION ABOUT A FIXED AXIS The moment of inertia I of a body composed of N particles is |
Moment of inertia |

| (9.6)

where m is the mass of a particle and r is the perpendicular distance of the particle from the axis of rotation.
For a rigid body rotating about a fixed axis, Newton’s second law for rotational motion is
Newton’s second law for rotational motion |

| (9.7)

where is the net external torque applied to the body, I is the moment of inertia of the body, and is its angular acceleration.
| 9.5 ROTATIONAL WORK AND ENERGY The rotational work WR done by a constant torque in turning a rigid body through an angle is |
Rotational work |

| (9.8)

The rotational kinetic energy KER of a rigid object rotating with an angular speed about a fixed axis and having a moment of inertia I is
Rotational kinetic energy |

| (9.9)

The total mechanical energy E of a rigid body is the sum of its translational kinetic energy , its rotational kinetic energy , and its gravitational potential energy (mgh):
Total mechanical energy |

where m is the mass of the object, v is the translational speed of its center of mass, I is its moment of inertia about an axis through the center of mass, is its angular speed, and h is the height of the object’s center of mass relative to an arbitrary zero level.
Conservation of total mechanical energy | The total mechanical energy is conserved if the net work done by external nonconservative forces and external torques is zero. When the total mechanical energy is conserved, the final total mechanical energy Ef equals the initial total mechanical energy E0: Ef = E0. |
| 9.6 ANGULAR MOMENTUM The angular momentum of a rigid body rotating with an angular velocity about a fixed axis and having a moment of inertia I with respect to that axis is |
Angular momentum |

| (9.10)

Conservation of angular momentum | The principle of conservation of angular momentum states that the total angular momentum of a system remains constant (is conserved) if the net average external torque acting on the system is zero. When the total angular momentum is conserved, the final angular momentum Lf equals the initial angular momentum L0: Lf = L0. |
Use Self-Assessment Test 9.2 to evaluate your understanding of Sections 9.4–9.6.

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