Chapter 8 Lecture Pearson Physics Rotational Motion and Equilibrium Prepared by Chris Chiaverina
Chapter Contents Describing Angular Motion Rolling Motion and the Moment of Inertia Torque Static Equilibrium
Describing Angular Motion Rotation is a part of everyday life. We live on a planet that rotates and revolves around the Sun. Automobiles have parts that rotate, as do devices that play CDs and DVDs. To gain a better understanding of rotational motion, we begin by considering the position, speed, and acceleration of a rotating object. A coordinate system with an origin is used to describe the motion of an object moving in a straight line. The same thing is done to describe the motion of a rotating object.
Describing Angular Motion The bicycle wheel shown in the figure below is free to rotate about its axle. The axle is the axis of rotation for the wheel. As the wheel rotates, every point on the wheel moves in a circular path centered about the axis of rotation.
Describing Angular Motion The location of a red spot on the bicycle tire in the figure is described by its angular position, that is, the angle θ that it makes with a given reference line. The reference line defines where the angular position is zero, θ = 0. The sign of angular position depends on its orientation relative to the reference line. Counterclockwise rotation from the reference line corresponds to positive angles, θ > 0. Clockwise rotation from the reference line corresponds to negative angles, θ < 0.
Describing Angular Motion While common units for measuring angles are the degree ( ) and revolution (rev), the most convenient unit for angle measurements in scientific calculations is the radian (rad). As the figure below indicates, the radian is the angle for which the length of a circular arc is equal to the radius of the circle.
Describing Angular Motion A comparison between angles measured in degrees and in radians is shown in the figure below. Angles are indicated around the circumference of the circle in both degrees and radians.
Describing Angular Motion As is illustrated in the following figure, the radian is useful because it provides a simple relationship between angles and arc lengths.
Describing Angular Motion Multiplying the angle θ (measured in radians) by the radius gives the arc length s: s = rθ. This relationship does not hold for degrees or revolutions. Since the arc length corresponding to the circumference of a circle equals 2πr, it follows from the relationship s = rθ that a complete revolution corresponds to θ = 2π radians. Therefore, 1 rev = 360 = 2π rad
Describing Angular Motion The units for angles radians, degrees, and revolutions are all dimensionless. In the relation s = rθ, for example, the arc length (s) and the radius (r) both have the SI unit of meter; hence the angle has no dimensions.
Describing Angular Motion As the figure below indicates, as the bicycle wheel rotates, the angular position of the spot of red paint changes. The angular displacement of the spot, Δθ, is defined as the difference between its final angle and its initial angle: Δθ = θ f θ i
Describing Angular Motion The angular displacement divided by the time during which the displacement occurs is the average angular velocity, ω av.
Describing Angular Motion Notice that ω is called the angular velocity, not the angular speed. This is because ω can be positive or negative, depending on the direction of rotation. Counterclockwise rotation corresponds to positive angular velocity, ω > 0. Clockwise rotation corresponds to negative angular velocity, ω < 0.
Describing Angular Motion The sign of ω indicates the direction of angular velocity, as shown in the figure below. The magnitude of the angular velocity is called the angular speed. The following example illustrates how the angular velocity is calculated.
Describing Angular Motion The angular and linear velocities of an object moving in a circle are related. At any instant, an object moving in a circle is also moving in a direction tangent to the circular path.
Describing Angular Motion The object's tangential speed is equal to the circumference of the circular path, d = 2πr, divided by the time required to complete one circuit, t = T. This gives the following: v t = d/t = 2πr/T = r(2π/t) Since 2π/T is the angular velocity ω, the tangential speed of a rotating object may be expressed as follows:
Describing Angular Motion If the angular velocity of a rotating object increases or decreases with time, then we say that the object has an angular acceleration. The average angular acceleration, α av, is the change in angular velocity, Δω, in a given time interval, Δt.
Describing Angular Motion The sign of the angular acceleration is determined by whether the change in angular velocity is positive or negative. If ω and α have the same sign, then the speed of rotation is increasing. If ω and α have opposite signs, then the speed of rotation is decreasing. These relationships are summarized in the following figure.
Describing Angular Motion The following example illustrates an application of the definition of average angular acceleration. Just as linear speed is equal to the radius times the angular speed, the linear acceleration is equal to the radius times the angular acceleration: tangential acceleration = radius x angular acceleration α t = rα
Rolling Motion and the Moment of Inertia As a bicycle wheel rolls freely, with no slipping between the tire and the ground, the wheel rotates about its axle, and at the same time the axle moves in a straight line. Thus, the rolling motion of a wheel is a combination of both rotational motion and linear motion.
Rolling Motion and the Moment of Inertia Because the forward motion of the axle exactly cancels out the backward motion of the bottom of the wheel, the instantaneous speed of the bottom of the wheel is zero. The top of the wheel has twice the speed of the axle. Thus, if a car's speed is v, as read on the speedometer, then the tops of its wheels have the speed 2v.
Rolling Motion and the Moment of Inertia Due to its inertia, an object at rest or moving in a straight line resists changes to that motion. A similar relationship occurs with rotational motion. Each object has a moment of inertia, I, which determines how easy or hard it is to change its rotation. An object with a large moment of inertia is difficult to start or stop rotating. For example, a merry-go-round has a large moment of inertia, whereas a baseball has a relatively small moment of inertia.
Rolling Motion and the Moment of Inertia Experiments show that an object's moment of inertia depends linearly on the mass of the object and on the square of the distance to the object from the axis of rotation. For a mass m at a distance r from the axis of rotation, the moment of inertia is I = mr 2.
Rolling Motion and the Moment of Inertia In a system with several particles with different masses and at different distances from the axis of rotation, the moment of inertia is the sum of the mr 2 terms for each object.
Rolling Motion and the Moment of Inertia The preceding results are valid for individual particles or groups of particles. What about a solid object, like a disk or sphere? The table below contains the moment of inertia for several uniform, rigid objects of various shapes and total mass m.
Rolling Motion and the Moment of Inertia For linear motion, the kinetic energy of a mass m moving with a speed v is. To find the kinetic energy of a rotating object, we replace the linear speed v with the angular speed ω and the mass m with the moment of inertia I, which we can think of as rotational mass. Thus the rotational kinetic energy of an object is one-half the product of the moment of inertia and the square of the angular speed.
Rolling Motion and the Moment of Inertia Thus the rotational kinetic energy of an object is one-half the product of the moment of inertia and the square of the angular speed.
Rolling Motion and the Moment of Inertia Connections between linear and rotational quantities such as these are summarized in the table below. We can apply these connections to momentum. By replacing mass with the moment of inertia and linear speed with angular speed, we obtain the angular momentum, L.
Torque A force can cause a rotation. The effectiveness of the force depends on where the force is applied and in which direction. In the figure below, we see that the force required far from the nut is much less than the force required near the nut. Less force is needed to open a revolving door if the force is applied far from the door's axis of rotation. r 1 F 1 r 2 r F 1 1 r 2 F 2 F 2 (a) (b)
Torque Thus the effectiveness of a force in causing a rotation depends on both the magnitude of the force and the distance from the axis of rotation to the force. Taking this information into account, we define a physical quantity called torque. Torque,, is the product of force and distance.
Torque
Torque The figure on the left below shows that a force acting in a direction that is directly toward or away from the axis of rotation will cause no rotation. That is, radial forces produce zero torque. The figure on the right below shows that if the force is at an angle θ relative to the radial line, then only the tangential component of the force will produce a torque.
Torque Therefore, the torque in the general case is distance times the tangential component of the force, rf cosθ:
Torque The figure below shows how torque can be defined in terms of a quantity known as a moment arm. The perpendicular distance from the axis of rotation to the line of the force is defined as the moment arm. From the figure, the length of the moment arm is r = r sinθ. Referring to the general equation for torque,
Torque The sign associated with a torque is determined by the type of angular rotation it would produce. A torque that causes a counterclockwise rotation is defined to be positive. A torque that causes a clockwise rotation is defined to be negative.
Torque The figure below illustrates a system with more than one torque. The net torque acting on the system is the sum of the individual torques, taking into account the appropriate sign for each.
Torque We can once again draw on the connections between linear and rotational quantities when considering the effect of torque on an object. Replacing the force with torque,, and the linear acceleration with angular acceleration, α, we obtain the rotational version of Newton's second law:
Torque Physical quantities that are conserved, such as energy and linear momentum, play an important role in physics. Angular momentum is another quantity that is conserved. Angular momentum is conserved when the net torque is zero. The motion of an ice skater may be used to illustrate how angular momentum is conserved. In the following figure, the skater pulls in her arms, reducing her momentum of inertia. However, since her angular momentum must stay the same, her angular speed must increase.
Torque In general, a large I and a small ω give the same angular momentum as a small I and a large ω: Iω = Iω
Static Equilibrium When an object is at rest and remaining at rest, it is said to be in a state of static equilibrium. The conditions for static equilibrium are the following: The total force acting on the object must be zero. This ensures that there is no linear acceleration. The total torque acting on the object must also be zero. This ensures that there is no angular acceleration.
Static Equilibrium An application of the these two conditions to the plank that supports the child in the figure below may be used to calculate the forces F 1 and F 2 acting on the plank. 3L/ 4 L/ 4 F 1 F 2 Axis of rotation y mg x The total force acting on the plank must be zero; therefore, F 1 + F 2 mg = 0
Static Equilibrium The total torque acting on the plank may be found by choosing the left end of the plank to be the axis of rotation. With this choice: Force F 1 exerts zero torque. Force F 2 produces a torque of +F 2 L since it acts at the far end of the plank in the counterclockwise direction. The weight of the child mg exerts a torque of mg since it acts in the clockwise direction at a distance from the axis.
Static Equilibrium Setting the total torques equal to zero, we find 0 + F 2 L mg(¾l) = 0 Therefore, and. In this example, the left end of the plank was arbitrarily selected as the axis of rotation. In general, you are free to chose an axis of rotation that is most convenient for a given situation.
Static Equilibrium The center of mass (CM) is the point at which an object can be balanced. In many ways, an object behaves as if all of its mass were concentrated at its center of mass. The location of the center of mass for two objects may be calculated using the following simple equation.
Static Equilibrium The figure below shows the center of mass of two objects. The center of mass is closer to the larger mass, or equidistant between the masses if they are equal. The center of mass (CM) is closer to the more massive object. m 1 m 2 CM m 1 m 2 CM When an object is supported at its center of mass, it is in static equilibrium since the torques acting on the object are balanced. m 1 m 2 CM
Static Equilibrium When the center of mass of an object is directly below the suspension point, the torque due to gravity is zero. This is because the force of gravity acts through the axis of rotation. Therefore, if you allow an object with any shape to hang freely, its center of mass will be directly below its suspension point.
Static Equilibrium The center of mass of an irregularly shaped object, such as a piece of wood cut into the shape of the continental United States, may be located by suspending the model from two or more points. The center of mass lies at the intersection of vertical lines extending downward from the suspension points.