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A PHYSICS GUIDE FOR… CHAPTER 9 NAME DATE PERIOD 9.1 Impulse and Momentum Ancient Babylonians described fractions of a circle in terms of 360 divisions that are now called degrees. A more modern method is based on the length of the radius and the distance around the circumference of the circle. If the object is spinning faster and faster, or spinning slower and slower, in a certain period of time, the object has an angular acceleration, . The relationship between linear and angular acceleration is a = r. In one spin, a point on the edge of a circle travels a distance equal to 2 times the radius of the circle. This defines the modern angular measurement of 2 radians = 360°. The unit of radians is added for “cosmetic purposes” to provide a unit analogous to the degree. The unit of one radian is equivalent to 360° / 2, or 1 rad = 57.3°. Although the point’s linear velocity works out to be its tangential speed, the point’s linear acceleration is not the centripetal acceleration in this case, since centripetal acceleration applies only to uniform circular motion, where the spin is not increasing or decreasing. As an object rotates, the change in the angle that it rotates is the angular displacement, . How far the point on the edge of the circle moves while the object spins through the angular displacement is given by the relationship d = r. The time it takes to rotate through that angular displacement is the angular speed, . Angular speed is defined by the rate change of angular displacement, or = /t. The relationship between the point’s tangential velocity and angular speed is vt = r. The linear speed of the point works out to be the tangential velocity, because its velocity vector can only be tangent to the circle. v=r = t vt =2 r T Given. Given. Substitution If is one complete spin, 2 rad, then t is the period T. EQUATION SUMMATION Displacement d = r Velocity = vt = r t Acceleration = a=r t vt =2 r ac = 42 r ac = vt2 2 T T r ANALOGOUS RELATIONSIPS LINEAR ANGULAR v f = vi + a t f = i + a t d = ½ (v f + vi) t = ½ ( f + i) t 2 d = vi t + ½ a t = i t + ½ t2 v f2 = vi2 + 2 a d f2 = i2 + 2 (1) (2) (3) (4) A PHYSICS GUIDE FOR… CHAPTER 9 NAME DATE PERIOD 9.2 Conservation of Momentum If linear displacements, speeds, and accelerations have analogous angular quantities, then it makes sense that force does as well. Torque is a force that tends to cause rotation around a point called the fulcrum. The amount of applied torque depends on the amount of force, the distance the force is from the fulcrum, and the angle in which the force is applied compared to the distance. There are two ways to express torque. = F r sin and = r F sin The quantity r sin is called the lever arm. It is the perpendicular distance from the fulcrum to the force. The quantity F sin is the component of the force perpendicular to the applied object. This is a subtle difference in the application alone. The preferred way to express and think of torques is = r F sin, since it is the perpendicular component of the force that tends to cause the rotation. The mathematics is still the same. F is the force, r is the distance from the fulcrum to the force, and is the angle between the two. The direction of the torque can also be confusing. The simplest description is to call the torque positive if it would tend to cause counterclockwise motion around the fulcrum. The torque is negative if it would tend to cause clockwise motion around the fulcrum. The net torque is found by adding all the torques, clockwise and counterclockwise (positive and negative). If the object is not spinning, the net torque will be equal to zero. A more convenient way to express that concept is to equate the sum of the clockwise torques to the sum of the counterclockwise torques net = 0 or cw net = ccw net If the object is spinning, then the net torque will be equal to the product of the object’s rotational inertia and the object’s rotational acceleration. net = I The object’s rotational inertia, also called the moment of inertia, indicates how the object responds to changes in rotational motion. This is the object’s rotation equivalent to its linear inertia. The higher the moment of inertia, the more difficult is is to start the object spinning, to spin it faster or slower, and to stop the object spinning. The amount of moment of inertia an object has depends on the distribution of the mass around the axis of rotation. If the axis of rotation changes, the moment of inertia changes. If the mass gets closer to or farther from the axis of rotation, the moment of inertia changes. Obviously, if the amount of mass changes, the moment of inertia changes. The moment of inertia for different objects spinning around different axes is listed below. To use these in the previous equation for net torque, select the correct object and axis for the situation. If the object or axis is not listed, the situation cannot be solved (Well, it can. The correct equation for the moment of inertia needs to be found, then proceed as needed.) Thin ring; axis through center: I = m r2 Thick ring; axis through center: I = ½ m (r12 + r22) Disk; axis through center: I = ½ m r2 Disk; axis tangent to edge: I = ¼ m r2 Solid ball; axis through center: (i.e. it is rotating): I = ⅖ m r2 Solid ball; axis at a distance, r, from object (i.e. it is revolving): I = m r2 Thin rod; length l; axis through center: I = m l2 Thin rod; length l; axis at one end: I = ⅓ m l2 Rectangular sheet; axis through center: I = ½ m (a2 + b2) A PHYSICS GUIDE FOR… CHAPTER 9 NAME DATE PERIOD 9.2 continued Newton’s Law of Inertia describes two states of equilibrium: static equilibrium (without motion) and dynamic equilibrium (with motion). If an object is in dynamic equilibrium, it can either move in a straight line at a constant speed (Fnet = 0) or it can spin at a uniform rate (cw net = ccw net). In addition to two states of equilibrium there exists two conditions of equilibrium. The first condition is translational equilibrium, in which the object is moving in a straight line at a constant speed (Fnet = 0). The second condition is rotational equilibrium, in which the object is spinning at a uniform rate (cw net = ccw net). An object in static equilibrium must meet both conditions of equilibrium. An object in motion can move in a straight line at a constant speed, or it can it can spin at a uniform rate, or both, or neither. If an object is spinning while in equilibrium, its axis of rotation will go through the center of mass. The center of mass is the point at which all the mass of the object can be considered to be concentrated without changing the object’s linear inertia. A net force applied through the center of mass from any direction will cause the object to accelerate without changing the rotational motion. Some objects in equilibrium are more stable than others. An object is stable if a net external force is needed to tip the object. The degree of stability depends upon the orientation of the object. The object remains stable if its center of mass remains either above or below the object’s base. When the tipping force is removed, the object returns to its stable position. If the object can be positioned to lower its center of mass or to have a wider base, the object will have greater stability. An object is neutrally stable if it may be overturned without changing raising or lowering its center of mass. Like a ball, tire, or tin can rolling across the floor. An object is unstable if it may be overturned easily, usually in response to a gravitational force, since the center of mass does not remain precisely over the base, allowing the center of mass to freely move to a lower position of greater stability. The center of mass for any object may be found by hanging the object by a string with one end of the string acting as a plumb line. The center of mass will lie along that line. Hanging the object from a second point on the object will mark the center of mass at the intersections of the two plumb lines. The object can be supported from either above or below its center of mass and not rotate. A person’s center of mass lies in the intersection of three planes: a vertical plane bisecting the body right from left; another vertical plane intersecting the ankles dividing the body front from back; a third plane horizontally dividing the body just above the male’s hip bone or the female’s hip joint. The path of the projectile is parabolic in shape. If the projectile is spinning, it will spin around its center of mass while its center of mass follows the parabolic path of the projectile.