Physics 4A Chapters 10: Rotation It is never to late to be what you might have been. George Eliot Only those who risk going too far can possibly find out how far they can go. T.S. Eliot Reading: pages 241 265 Outline: angular quantities position and displacement velocity and acceleration angular quantities as vectors (read section 10-3 on your own) equations of constant acceleration relating linear and angular quantities position and speed tangential and radial acceleration torque and rotational inertia torque rotational inertia (moment of inertia I) Newton s 2 nd law for rotation Determining I / parallel-axis theorem equations of constant acceleration work and rotational KE rotational KE work-kinetic energy theorem power Problem Solving Many problems of this chapter are mathematically identical to those of one-dimensional linear kinematics, discussed in Chapter 2: θ replaces x, w replaces v, and α replaces a. If a problem involves rotation with constant angular acceleration, first write the kinematic equations θ (t) = θ 0 + ω 0 t + ½ α t 2 and ω(t) = ω 0 + α t, then solve them simultaneously for the unknowns. As an alternative, search Table 10-1 for the equation containing the quantities that are given in the problem statement. Recall that when you solve one-dimensional problems you can always place the origin at any point you choose. In particular you might select the origin to be at the position of the particle when time t = 0 and thus take x 0 to be zero. Similarly, to describe rotational motion you can always orient the fixed reference frame and select the reference line in the body so the initial
angular position θ 0 is zero. This is usually helpful because it reduces the number of algebraic symbols you must carry in the calculation. Some problems deal with relationships between linear and angular variables. Given the radius of the orbit of a point in a rotating body you should be able to obtain values for the linear variables from values for the angular variables and vice versa. Use Δs = rδθ, v = rω, a t = rα, and a r = rω 2. Don't forget that the angular variables must be expressed in radian measure. This may require a change in units. If you are asked to calculate the acceleration of a point in a spinning body, remember it has both tangential and radial components. You need to know ω to compute the radial component (a r = rω 2 ) and α to compute the tangential component (a t = rα). These might be given or might be solutions to another problem. If you are asked for the magnitude of the acceleration, use 2 2 a = a + a. t r You should know how to compute the rotational inertia of a system composed of a small number of discrete particles. Given the masses and positions of the particles, you must evaluate the sum I = 2 mr i i, where r i is the perpendicular distance of particle i from the axis of rotation. You may need to use some geometry to obtain r i. If, for example, the z coordinate axis is the axis of rotation, then r 2 i = x 2 i + y 2 i. For many problems you will want to make use of Table 10-2. For some you will need to use the parallel-axis theorem. Some problems ask you to compute the torque associated with a given force. Simply find the force component tangent to the circular orbit at the application point, then multiply by the perpendicular distance from the rotation axis to the application point. For fixed axis rotation you may treat the torque as a scalar. Assign a sign to it according to whether it tends to cause rotation in the counterclockwise (+) or clockwise (-) direction when it is applied to an object at rest. Many problems deal with the rotational motion of a rigid body about a fixed axis and can be solved using τ net = Iα. Draw a force diagram. This is a sketch of the rotating object with the rotation axis perpendicular to the page. Draw arrows for all forces acting on the object, with their tails at the points of application. Label the arrows with symbols for the force magnitudes. Also label the distances from the rotation axis to the points of application. Write the sum of the torques in terms of the force magnitudes and the distances to the rotation axis. Be sure to include the signs of the individual torques (positive for counterclockwise and negative for clockwise, for example). Set the sum of torques equal to Iα, identify the known quantities and solve for the unknown quantities. Some problems involve several bodies, some in translation and some in rotation. The accelerations of the translating bodies and the angular accelerations of rotating bodies are usually related. Perhaps a string runs from a translating body and around a rotating body. Then, if the string does not slip or stretch, the magnitude of the tangential acceleration of a point on the rim of the rotating body must be the same as the magnitude of the acceleration of the translating body. You write a = ±αr. Which sign you use depends on which directions you chose for positive acceleration and positive angular acceleration. If you use the + sign, you are saying that a positive acceleration of the translating body is consistent with a positive angular acceleration of the rotating body. If you use the - sign, you are saying a positive acceleration of the translating body is consistent with a negative angular acceleration of the rotating body.
To solve this type problem, draw a force diagram for each body. For a rotating body, choose the direction of positive rotation (the counterclockwise direction, say) and write the sum of the torques, taking their directions into account. Set the sum equal to Iα. For a translating body, choose a coordinate system with one axis in the direction of the acceleration, if possible. Write the sum of the force components for each coordinate direction and set each sum equal to the product of the mass of the body and the appropriate acceleration component. Write all equations in terms of the magnitudes of the forces and appropriate angles. Use algebraic symbols, not numbers. Don't forget tension forces of strings, gravitational forces, normal forces, and frictional forces, if they act. If a string is wrapped around a disk that is free to rotate and if the string does not slip on the disk, then the force exerted by the string on the disk is the tension force T. Since the string must be tangent to the disk, the torque it exerts has magnitude TR, where R is the radius of the disk. If a string passes over a pulley, then the tension force of the string might be different on different sides. If T 1 is the tension force on one side and T 2 is the tension force on the other side, then ±(T 1 - T 2 )R is the net torque exerted by the string on the pulley. The sign, of course, depends on which direction of rotation is taken to be positive. The second-law equations (for translation and rotation) and the equation or equations that link the accelerations of translating bodies and the angular accelerations of rotating bodies are solved simultaneously for the unknowns. Some problems can be solved using the work-kinetic energy theorem for rotation or the principle of energy conservation. Consider using an energy technique if angular displacements and angular velocities are given or requested. On the other hand, use Newton's second law for rotation if angular accelerations and torques or forces are given or requested. If you use an energy method, write the initial and final kinetic energies for some time interval in terms of given or requested quantities. These are usually the rotational inertia and angular velocities. If the system also includes a translating object, such as a falling body that exerts a torque (via a string) on a rotating object, you must include the kinetic energy of that body. Then write an expression for the total work done by external forces and set the expression equal to the change in the total kinetic energy. Identify the known quantities and solve for the unknown quantities. Questions and Example Problems from Chapter 10 Question 1 The figure below is a graph of the angular velocity of a rotating disk. What are the (a) initial and (b) final directions of rotation? (c) Does the disk momentarily stop? (d) Is the angular acceleration positive or negative? (e) Is the angular acceleration constant or varying?
Question 2 The figure below is a graph of the angular velocity versus time for a disk rotating like a merrygo-round. For a point on the rim of the disk, rank the four instants a, b, c, and d according to the magnitude of (a) the tangential acceleration and (b) the radial acceleration, greatest first. Question 3 The figure below shows an assembly of three small spheres of the same mass that are attached to a massless rod with the indicated spacings. Consider the rotational inertia I of the assembly about each sphere, in turn. Then rank the spheres according to the rotational inertia about them, greatest first. Problem 1 The angular position of a point on a rotating wheel is given by θ = 2.0 + 4.0t 2 + 2.0t 3, where θ is in radians and t is in seconds. At t = 0, what are (a) the point s angular position and (b) it s angular velocity? (b) What is its angular velocity at t = 4.0 s? (d) Calculate its angular acceleration at t = 2.0 s. (e) Is its angular acceleration constant?
Problem 2 A wheel rotating about a fixed axis through its center has a constant angular acceleration of 3.0 rad/s 2. During a certain 4.0 s interval, the wheel turns through an angle of 120 rad. Assuming that the wheel started from rest, how long has it been in motion at the start of the 4.0 s interval? Problem 3 A merry-go-round rotates from rest with an angular acceleration of 1.50 rad/s 2. How long does it take to rotate through (a) the first 2.00 rev and (b) the next 2.00 rev? Problem 4 A vinyl record on a turntable rotates at 33.3 rev/min. (a) What is its angular speed in radians per second? What is the linear speed of a point on the record at the needle when the needle is (b) 15 cm and (c) 7.4 cm from the turntable axis?
Problem 5 An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10 m and, in starting, rotates according to θ = 0.30t 2, where t is in seconds and θ is in radians. When t = 5.0 s, what are the magnitudes of the astronaut s (a) angular velocity, (b) linear velocity, (c) tangential acceleration, and (d) radial acceleration? Problem 6 The figure below shows three 0.01000 kg particles that have been glued to a rod of length L = 6.00 cm and negligible mass. The assembly can rotate around a perpendicular axis through point O at the left end. If we remove one particle (that is, 33% of the mass), by what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one?
Problem 7 The uniform solid block in the figure below has mass 0.172 kg and edge lengths a = 3.5 cm, b = 8.4 cm, and c = 1.4 cm. Calculate its rotational inertia about an axis through one corner and perpendicular to the large faces. Problem 8 The body in the figure below is pivoted at O. Three forces act on it: F A = 10 N at point A, 8.0 m from O; F B = 16 N at B, 4.0 m from O; and F C = 19 N at C, 3.0 m from O. What is the net torque about O?
Problem 9 The figure below shows particles 1 and 2, each of mass m, attached to the ends of a rigid massless rod of length L 1 + L 2, with L 1 = 20 cm and L 2 = 80 cm. The rod is held horizontally on the fulcrum and then released. What are the magnitudes of the initial accelerations of (a) particle 1 and (b) particle 2? Problem 10 A wheel of radius 0.20 m is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is 0.050 kg m 2. A massless cord wrapped around the wheel is attached to a 2.0 kg block that slides on a horizontal frictionless surface. If a horizontal force of magnitude P = 3.0 N is applied to the block as shown in the figure below, what is the magnitude of the angular acceleration of the wheel? Assume the string does not slip on the wheel.
Problem 11 In the figure below, two blocks of mass m 1 = 400 g and m 2 = 600 g, are connected by a massless cord that is wrapped around a uniform disk of mass M = 500 g and radius R = 12.0 cm. The disk can rotate without friction about a fixed horizontal axis through its center; the cord cannot slip on the disk. The system is released from rest. Find (a) the magnitude of the acceleration of the blocks. (b) the tension T 1 in the cord at the left, and (c) the tension T 2 in the cord at the right. Use energy considerations to find the speed of the blocks after the 600 g block has fallen 25 cm.
Problem 12 A uniform spherical shell of mass M = 4.5 kg and radius R = 8.5 cm can rotate about a vertical axis on frictionless bearings. A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 3.0 10-3 kg m 2 and radius r = 5.0 cm, and is attached to a small object of mass m = 0.60 kg. There is no friction on the pulley s axle; the cord does not slip on the pulley. What is the speed of the object when it has fallen 82 cm after being released from rest? Use energy considerations. Problem 13 A meter stick is held vertically with one end on the floor and is then allowed to fall. Find the speed of the other end when it hits the floor, assuming that the end on the floor does not slip. (Hint: Consider the stick to be a thin rod and use the conservation of energy principle.)