Chapter 8

• In purely rotational motion, all points on the object move in circles around the axis of rotation r.
• The radius of the circle in purely rotational motion is r.
• All points on a straight line drawn through the axis of rotation in purely rotational motion move through the same angle in the same time.
• The angle θ in radians is defined as where l is the arc length.
• An object that is rotating has rotational kinetic energy.
• If an object is translating as well, the translational kinetic energy must be added to the rotational to find the total kinetic energy.
• The angular acceleration is proportional to the torque and inversely proportional to the rotational inertia.
• If the net torque on an object is zero, its angular momentum does not change.
• Angular momentum is L = I ω.
• Angular displacement is defined as Δθ = θ2 − θ1.
• The average angular velocity is defined as the total angular displacement divided by time.
• The instantaneous angular velocity is defined as the angular velocity at a specific time.
• The angular acceleration is the rate at which the angular velocity changes with time.
• Each point on a rotating body has an angular velocity ω and a linear velocity v.
• Objects farther from the axis of rotation will move faster.
• If the angular velocity of a rotating object changes, it has a tangential acceleration.
• Even if the angular velocity is constant, each point on the object has a centripetal acceleration.
• The frequency is the number of complete revolutions per second: f = ω/2π.
• Frequencies are measured in hertz: 1 Hz = 1 s−1.
• The period is the time one revolution takes.
• The torque is defined as: (8-10a)
• A longer lever arm is very helpful in rotating objects.
• To make an object start rotating, a force is needed; the position and direction of the force matter as well.
• Knowing that F = ma, we see that τ = mr 2 α.
• The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.
• The distribution of mass matters here—these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.
• The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation—compare (f) and (g), for example.
• The quantity I = Σ mr 2 is called the rotational inertia of an object.
• To solve problems in rotational dynamics, draw a diagram, decide what the system comprises, draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act, find the axis of rotation; calculate the torques around it, apply Newton’s second law for rotation, apply Newton’s second law for translation and other laws and principles as needed, and solve.
• The relationship between linear and angular speeds is v = r ω.
• Check your answer for units and correct order of magnitude.
• If the net torque on an object is zero, the total angular momentum is constant: I ω = I0 ω0 = constant.
• The total torque on an object can be written as the rate of change of angular momentum.
• Angular acceleration and angular momentum vectors also point along the axis of rotation.
• Angles are measured in radians; a whole circle is 2π radians.
• Angular acceleration is the rate of change of angular velocity.
• In analogy with linear momentum, we can define angular momentum L: © 2014 Pearson Education, Inc. (8-18)
• The equations for rotational motion with constant angular acceleration have the same form as those for linear motion with constant acceleration.
• When using conservation of energy, both rotational and translational kinetic energy must be taken into account.
• Angular velocity is the rate of change of angular position.