PHYS 220 Exam I Study Guide
PHYS 220 Exam I Study Guide Phys 220
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This 11 page Study Guide was uploaded by Kathryn Chaffee on Thursday February 4, 2016. The Study Guide belongs to Phys 220 at Purdue University taught by Dr. Elliot in Fall 2015. Since its upload, it has received 45 views. For similar materials see Physics 220 in Physics 2 at Purdue University.
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Date Created: 02/04/16
PHYSICS 220 EXAM I Chapter 1: Kinematics: Motion in One Dimension Motion is a change in an object’s position relative to a given observer during a certain change in time. Physicists say motion is relative, meaning that the motion of any object of interest depends on the point of view of the observer. A reference frame includes an object of reference, a coordinate system with a scale for measuring distances, and a clock to measure time. Linear motion is a model of motion that assumes that an object, considered as a point-like object, moves along a straight line. We can represent motion in more detail by adding velocity arrows to each dot that indicate which way the object is moving and how fast it is moving. The longer the arrow, the faster the motion. The small arrow above the letter v indicates that this characteristic of motion has a direction and a magnitude – called a vector quantity. The ν doesn’t tell us the exact increase or decrease in the velocity; it only indicates a qualitative difference between the velocities at two adjacent points in the diagram. A motion diagram helps represent motion qualitatively. Time: a clock reading Time interval: a difference in clock readings Time and time interval are both scalar quantities. The position of an object is its location with respect to a particular coordinate system. The displacement of an object is a vector that starts from an object’s initial position and ends at its final position. The magnitude of the displacement vector is called distance. The path length is how far the object moved as it travelled from its initial position to its final position. The vector that points from the initial position to the final position is the displacement vector. Kinematics means description of motion. Always keep in mind that representations of motion (motion diagrams, tables, kinematics graphs, ect.) depend on the reference frame. Together, speed and direction are called velocity, and this is what the slope of a position-versus-time graph represents. Velocity and speed for constant velocity linear motion: x −x v = 2 1= ∆ x x 2 −t1 ∆t This equation can also be used to determine the average velocity, which is the ratio of the change in position and the time interval in which this change occurred. Speed is the magnitude of the velocity and is always a positive number. Scalar quantities only have magnitude. Vector quantities have magnitude and direction. Scalar Quantities Vector Quantities Time Displacement Time interval Velocity Position Acceleration Distance Force Speed Average speed Path length Position equation for constant velocity linear motion: x=x ov x The velocity of an object at a particular time is called the instantaneous velocity. For motion at constant velocity, the instantaneous and average velocities are equal; for motion with changing velocity, they are not. An object’s average acceleration during a time interval t is the change in its velocity v divided by that time interval: v −v ∆v a= 2 1 = t2−t1 ∆t If the change in time is very small, then the acceleration given by this equation is the instantaneous acceleration of the object. For one- dimensional motion, the component of the average acceleration along a particular axis is v2 x 1 x ∆v x ax= t −t = ∆t 2 1 It is possible for an object to have a zero velocity and a nonzero acceleration – for example, at the moment when an object starts moving from rest. An object can also have a nonzero velocity and zero acceleration – for example, an object moving at constant velocity. Rearranging the equation above we get an expression for the changing velocity of the object as a function of time: v =v +a t x ox x Position of an object during linear motion with constant acceleration: 1 2 x=x +o tox a x 2 Alternate equation for linear motion with constant acceleration: 2a x(x =v0)v x o x Galileo hypothesized that free fall occurs exactly the same way for all objects regardless of their mass and shape. g = 9.8 m/s 2 Chapter 2: Newtonian Mechanics A system is the object that we choose to analyze. Everything outside that system is called its environment and consists of objects that might interact with the system and affect its motion through external interactions. A force is a physical quantity that characterizes how hard and in what direction an external object pushes or pulls on the system object. Gravity represents the interaction of planet Earth with an object. Normal forces are contact forces. We have two possible ideas that related motion and force: 1) An object’s velocity always points in the direction of the sum of the forces that other objects exert on it 2) An object’s velocity change always points in the direction of the sum of the forces that other objects exert on it An inertial reference frame is one in which an observer sees that the velocity of the system object doesn’t change if no other object exerts forces on it or if the sum of all forces exerted on the system object is zero. For observers in noninertial reference frames, the velocity if the system can change even though the sum of forces exerted on it is zero. Newton’s First Law of Motion: For an observer in an inertial reference frame when no other objects exert forces on a system object or when the forces exerted on a system object add to zero, the object continues moving at constant velocity. Inertia is the phenomenon in which a system object continues to move at constant velocity when the sum of the forces exerted on it by other objects is zero. The acceleration is proportional to the sum of the forces. Mass characterizes the amount of matter in an object. When the same unbalanced force is exerted on two objects, the object with greater mass has a smaller acceleration. The unit of mass is called the kilogram. Mass is a scalar quantity. The acceleration is inversely proportional to the mass of the system. Newton’s Second Law: F a = ∑ onsystem system msystem The acceleration of a system object is proportional to the vector sum of all forces being exerted on the object and inversely proportional to the mass of the object. Gravitational force: The magnitude of the gravitational force that Earth exerts on any object when or near its surface equals the product of the object’s mass and the constant g F E onO gO Newton’s Third Law: When two objects interact, object 1 exerts a force on object 2. Object 2 in turn exerts an equal-magnitude, oppositely directed force on object 1 Fobject1onobjectobject2onobject1 Chapter 3: Applying Newton’s Laws We can always replace any force F with two perpendicular forces Fx and Fy, as long as the perpendicular forces graphically add to F. In this case, the perpendicular forces along the perpendicular x- and y-axes are called the x- and y-vector components of the original force F. Components of a vector quantity: Fx=±Fcosθ Fy=±Fsinθ Newton’s second law in component form: the acceleration of an object in the x-direction is the sum of the x-components of the forces exerted on it by its mass. The acceleration of the object in the y-direction is the sum of the y-components of the forces divided by its mass. Newton’s second law rewritten in scalar component form becomes: F1onSxF 2onSxF 3onSx… ax= m S F1onSyF 2onSyF 3onSy… ay= m S A scalar component is positive if it falls along a positive axis and negative if it falls along a negative axis. For two objects linked together: m 2 a= m 1m 2 Static friction force is parallel to the surfaces of two objects that are not moving in relation to each other and opposes the tendency of one object to move across the other. The static friction force changes magnitude to prevent motion – up to a maximum value. When the external force exceeds this static friction force, the object begins to move. This maximum resistive force that the surface can exert on the object is called the maximum static friction force. fsmaxμ s If the normal force exerted by a surface on an object increases, the maximum static friction force the surface exerts on the object increases proportionally. f This ratio μs= S max is called the coefficient of static friction μ s N for a particular pair of surfaces. The coefficient of static friction is a measure of the relative difficulty of sliding two surfaces across each other. The easier it is to slide one surface on the other, the smaller the value of μs . For a block that is already in motion, we find a similar relationship between the resistive friction force exerted by the surface on the block and the normal force exerted by the surface on the block. There are, however, two differences: 1) Under the same conditions, the magnitude of the kinetic friction force is always lower than the magnitude of the maximum static friction force 2) The resistive force exerted by the surface on the moving object doesn’t vary but has a constant value As with the static friction force f s the magnitude of this kinetic friction force f dkpends on the roughness of the contacting surfaces (indicated by a coefficient of kinetic friction μ ) and on the k magnitude N of the normal force exerted by one of the surfaces on the other, but not on the surface area of contact. The word kinetic indicates that the surfaces in contact are moving relative to each other. fk=μ k Projectiles are objects launched at an angle relative to a horizontal surface. We can analyze projectile motion by independently considering the ball’s vertical and horizontal motion. Earth exerts a gravitational force on the ball, so its upward speed decreases until it stops at the highest point, and then its downward speed increases until it returns to your hands (the vertical component of the ball’s acceleration is constant due to the gravitational force exerted on it by Earth). The ball also moves horizontally with respect to the floor. According to Newton’s first, the ball’s horizontal velocity does not change once it is released and is the same as your horizontal component of velocity. The horizontal and vertical motions of the ball are independent of each other. Projectile motion in the x-direction: ax=0 vx=v Ox coOθ v (¿¿Ocosθ)t x=x O¿ Projectile motion in the y-direction: a =−g y vy=v Oy t=y sinO+ (−g )t y=y + v(sinθ t−)gt 1 2 O O 2 Chapter 4: Circular Motion When an object is moving in a circle at a constant speed, its acceleration at any position points toward the center of the circle. This acceleration is called radial acceleration. Hypothesis: The sum of the forces exerted on an object moving at constant speed along a circular path points toward the center of that circle in the same direction as the object’s acceleration. The magnitude of radial acceleration is proportional to the speed squared. ar∝v 2 The magnitude of the radial component of the acceleration of an object moving in a circular path is inversely proportional to the radius of the circle. 1 ar∝ r The two proportionalities combined give the equation: v2 ar∝ r For an object moving at constant speed v on a circular path of radius r, the magnitude of the radial acceleration is 2 a = v r r 2 A large v in the numerator leads to a large acceleration. A small radius r in the denominator also leads to a large acceleration. The acceleration is zero if the radius in the denominator is infinite (equivalent to the object moving in a straight line). When an object repeatedly moves in a circle, we can describe its motion with another useful physical quantity, its period T. The period equals the time interval it takes an object to travel around the entire circular path one time and has the units of time. For constant speed circular motion we can determine the speed of an object by dividing the distance travelled in one period (the circumference of the circular path, 2πr) by the time interval T it took the object to travel that distance (its period), or v=2πr/T. Thus, T= 2πr v We can express the radial acceleration of the object in terms of its period by inserting this equation into the equation above: 4π r ar= 2 T If the speed of the object is extremely large, its period would be very short, thus, its radial acceleration would be very large. If the speed of the object is small, its period would be large and its radial acceleration would be small. Circular motion component form of Newton’s second law: For the radial direction (the axis pointing toward the center of the circular path), the component form of Newton’s second law is ΣF ar= r m or m a =ΣF r r For some situations (for example, a car moving around a highway curve or a person standing on the platform of a merry-go-round), we also include in the analysis the force components along a perpendicular vertical y-axis: m ay=ΣF =y When an object moves with uniform circular motion, both the y- component of its acceleration and the y-component of the net force exerted on it are zero. The gravitational force that Earth exerts on the Moon depends on the inverse square of its distance from the center of Earth: 1 FE onMoonatr2 r Newton assumed that the gravitational force of the Earth on the moon must be proportional to the Moon’s mass: FE onMm Moon If Earth exerts a gravitational force on the Moon, then according to Newton’s third law the Moon must also exert a gravitational force on Earth that is equal in magnitude. Because of this, Newton decided that the gravitational force exerted by Earth on the Moon should also be proportional to the mass of Earth: F =F ∝m E onM M onE Earth m m FE onMF M onE EarthMoon r2 −11 2 2 Universal gravitational constant: G=6.67x10 N ∙m /kg Newton’s law of universal gravitation: The magnitude of the attractive gravitational force that an object with m ex1rts on an object with m 2eparated by a center-to-center distance r is: F =G m1m 2 g1on2 r2 Kepler’s laws: 1) Kepler’s First Law of Planetary Motion – The orbits of all planets are ellipses with the Sun located at one of the ellipse’s foci 2) Kepler’s Second Law of Planetary Motion – When a planet travels in an orbit, an imaginary line connecting the planet and the Sun continually sweeps out the same area during the same time interval, independent of the planet’s distance from the Sun 3) Kepler’s Third Law of Planetary Motion – The square of the period T of the planet’s motion divided by the cube of the semi-major axis of the orbit (which is half the maximum diameter of an elliptical orbit or the radius r of a circular one) equals half the same constant for all the known planets: T 2 3=K r Chapter 5: Impulse and Linear Momentum Lavoisier defined an isolated system as a group of objects that interact with each other but not with external objects outside the system. Law of constancy of mass: when a system of objects is isolated (a closed container), its mass equals the sum of the masses of its components and does not change – it remains constant in time. When the system is not isolated, the mass might change. However, this change is not random – it is always equal to the amount of mass leaving or entering the system from the environment. Mass is called a conserved quantity. A conserved quantity is constant in an isolated system. When the system is not isolated, we can account for the changes in the conserved quantity by what is added to or subtracted from the system. The linear momentum of a single object is the product of its mass and velocity: p=mv Linear momentum is a vector quantity that points in the same direction as the object’s velocity. The SI unit of linear momentum is (kgm/s). The total linear momentum of a system containing multiple objects is the vector sum of the momenta of the individual objects. Momentum constancy of an isolated system: The momentum of an isolated system is constant. For an isolated two-object system: m v +m v =m v +m v 1 1i 2 2i 1 1 f 2 2 f Because momentum is a vector quantity and the equation above is a vector equation, we will work with its x- and y- component forms: m v +m v =m v +m v 1 1ix 2 2ix 1 1 fx 2 2 fx m v +m v =m v +m v 1 1iy 2 2iy 1 1 fy 2 2 fy The impulse of a force is the product of the average force F exerted av on an object during a time interval and that time interval: J=F (t −t )F ∆t av f i av Impulse is a vector quantity that points in the direction of the force. The impulse has a plus or minus sign depending on the orientation of the force relative to a coordinate axis. The SI unit for impulse is Ns, the same unit for momentum. Impulse-momentum equation for a single object: If several external objects exert forces on a single-object system during a time interval, the sum of their impulses causes a change in momentum of the system object: p fp =Σi=ΣF onSystemtf−t i The change in momentum of a system is equal to the net external impulse exerted on it. If the net impulse is zero, then the momentum of the system is constant. This idea, expressed mathematically as the generalized impulse-momentum principle, accounts for situations in which the system includes one or more objects and may or may not be isolated. The generalized impulse-momentum principle means that we can treat momentum as a conserved quantity. For a system containing one or more objects, the initial momentum of the system plus the sum of the impulses that external objects exert on the system during the time interval equals the final momentum of the system. m v +m v +… +ΣF t −t =(m v +m v +…) ( 1 1i 2 2i ) onSys( f i) 1 1 f 2 2 f When a system object collides with another object and stops, the system object travels what is called the stopping distance. By estimating the stopping distance of the system object, we can estimate the stopping time interval. Here’s how we can use a known stopping distance to estimate the stopping time interval: Assume that the acceleration of the object while stopping is constant. In that case, the average velocity of the object while stopping is just the sum of the initial and final velocities divided v fx ix by 2: v average x 2 Thus, the stopping displacement (x fx )i and the stopping time interval t −t are related by the kinematics equation ( f i (vfxv ix x fx =i average ft =i (tf−t i) 2 Rearrange this equation to determine the stopping time interval 2(x fx ) i tf−t i v +v fx ix The main idea behind the jet propulsion method is that when an object ejects some of its mass in one direction, it accelerates in the opposite direction.
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