Quantities needed to describe the motion of an object: Need to know how position changes with time

Transcription

1 Chapter 1 : Laws of Motion Quantities needed to describe the motion of an object: Need to know how position changes with time Position is location in space with respect to some other position The Reference position is called the origin O If the object is at a place P then it is described by a Distance ( in meters) from O and a direction :r and is a function of time It is a vector quantity: has magnitude and direction If you are moving, then in an interval of time t seconds You move through a distance r and your displacement is r This is shown in the next slide.

2 y axis Q rr r O r P x axis Speed is distance covered in time t which is : v r t Velocity is directed speed, for movement from P to Q: v r t, vector Note that t is just a number or a scalar

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4 If the speed is not changing then the object will keep moving in the same direction, i.e. the direction of PQ If your speed was, say, 10 meters per sec = 10 m sec Then you will continue move at this speed. Your velocity,v can change in three possible ways: (1) Direction remaining constant and speed increasing or decreasing (2) Speed remaining constant and direction changing (3) Both speed and direction changing To achieve any of the above changes, a mechanical force must be applied and change in velocity is along the direction of the applied force

5 If velocity is a constant, then the object will move with constant speed in the direction of the velocity vector. Concept of Inertia: It is the property of an object that describes the tendency of a body in motion to remain in motion or a body at rest to remain at rest, in the absence of presence of any external force on the body. First deduced by Galileo had to reduce effects of friction. Newton quantified it in his First Law of Motion: An object that is not subject to any outside forces moves at a constant velocity covering equal distances in equal times along a straight-line path. In ice skating, friction is minimal as there is a layer of water between the skate and the ice (effect of pressure affecting the melting temperature of ice) and the skater (or an ice puck) can coast for a long time without decreasing speed.

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7 Vector position can can also be defined by giving the angle that vector r makes with the x axis and its length. Definition of angle is shown on the right.

8 Rotation and motion in a circle(counter clockwise) the linear velocity vr Note is radians/sec and v is meters/sec v r ref axis Knowing as a function of timet Describes this motion as a function of time Rate of change of angle with time gives the angular velocity t the quantity is a vector whose arrow defines whether rotation is clockwise or anticlockwise

9 Newton's second law of motion can be considered either a definition of inertial mass or a definition of Force. Mechanical force is that quantity which causes a change in the velocity of the object on which it acts. Larger the force, greater the change in velocity. Larger the mass smaller the resulting acceleration (change in velocity) for the same applied force. As velocity is a vector, so is change in velocity or acceleration. For an object with mass m and an applied force F Newton's II Law is written as Fma Mass may be thought of as amount of substance in the object.

10 Fundamental and undefinable quantites in nature: In describing motion we deal with three basic quantities: Length, Mass and Time: L,M,T By a point we understand it as a place but it has no size! A length is made up of an infinite number of points each of which has no ' size', but length we can measure. Length is a fundamental quantity of nature we denote its dimensions as L and its units are defined in terms of a standard meter kept in vaults in Paris! Time is the interval between events occurring in nature. We know what it means but it cannot be defined in terms of anything else. Mass (or inertial mass) is the substance in a body. We understand what it means but we cannot define it in terms of something else.

11 Units of these fundamental Quantities Length is measured (in science) in terms of a standard length: Meter or m Time is measured in seconds or s Mass is measured in kilograms or kg Derived Quantities: speed or velocity is displacement per unit time: v r t length time Its dimentions are in terms of fundamental quantities: L T Its units are measured in m s Acceleration is change in velocity per unit time L T T L T 2L T 2 It is measured in units of m s 2 Force has dimensions from Fm a given by M L It is measured in units of kg m s 2 T 2

12 Instead of having to write for units of F kg m s 2 We define a Newton (in honor of Isaac Newton) as the force if mass is 1 kg and acceleration is 1 m/s^2. As an example take mass to be 50 kg, and acceleration to be 30 m per sec per sec the Force is F 50 x Newtons Downward acceleration due to gravity is g = 9.8 m s 2 If your mass is 70 kg then the gravitational force acting on you is Fm g70 x 9.8~ 700 Newtons

13 Other Derived quantities: Area = Length x Length = L^2 units = m^2 Volume = Length x Length x Length = L^3 units m^3 Work = Force x distance = (M L/ T^2) xl = M L^2 T^-2 units Joules Joule = 1 Newton x 1 meter or 1 Newton meter. Energy has same dimensions as Work Pressure is Force per unit area = (M L /T^2)/L^2 = (ML^2 T^-2)/L^3 = Joules/cubic meter.

14 Force of gravity near the Earth' s surface The first fundamental force investigated was that of gravity. Galileo made the first really quantitative study of motion of objects falling under the pull of the earth. He showed that all objects,irrespective of their mass fell down to the earth's surface with the same acceleration. This was shown in a legendary drop of different objects from the top of leaning tower of Pisa. He slowed down the fall under gravity using inclined planes to reduce the acceleration due to gravity so that he could measure time progress in motion using ' water clocks ' and pendulum clocks. He determined that acceleration due to gravity was ~ 10 m/s^2

15 Rolling eliminated frictional losses. He deduced the independence of motion along the horizontal and along the vertical directions. He showed that when F(net)=0 velocity was unchanged. He determined that: velocity increase vv 0 at distance coverd x 1 2 a t 2 More complete statement is: x(t)= x 0 v 0 t 1 2 a t 2

16 Mass and Weight: Weight is the force that a mass feels near the surface of the earth due to acceleration due to gravity. We call the acceleration due to gravity g Its numerical value is known to be 9.8 ms 2 Using Newton's II law we can write for the Force due to gravity on mass m as: Wm g Newtons As weight is proportional to mass all objects whatever their mass have the same acceleration according to: Fmamg and ag This is called the Principle of Equivalence and no deviation from it has been found up to now. It is also one of the basic assumptions of Einstein's theory of general relativity! This is true if there is no air resistance and if buoyancy is not present.

17 F2 on 1 F1 on 2 r M 1 M 2 Forces of Gravity between any two masses

18 Here I mention, Newton's Law of Universal Gravitation: There is a force between any two massive objects which is (1) always attractive (2) proportional to the product of the masses (3) proportional inversely to the square of the distance between the masses. For two masses m 1 and m 2 separated by distance r Gravitational force F g m m 1 2 G m M 1 2 r 2 r 2 the minus sign indicates attractive force And F1on 2 F2 on 1 magnitude of acceleration of 1 due to pull of 2 Using Newton's 2nd Law :Fma acceleration of 1 is : a 1 m 2 r 2 Does not depend on its own mass and acceleration of 2 is a 2 m 1 r 2