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Circular motion and Gravitational (chapter no 7) Centripetal Force: “The force which compels a body to move along a circular path” Whenever an object moves in a circular path we know the object is accelerating because the velocity is constantly changing direction. All accelerations are caused by a net force acting on an object. In the case of an object moving in a circular path, the net force is a special force called the centripetal force. Centripetal is Latin for "center seeking". So a centripetal force is a center seeking force which means that the force is always directed toward the center of the circle. Without this force, an object will simply continue moving in straight line motion Equation for centripetal force: centripetal force depends upon on following things. 1. Velocity of the body: centripetal force is directly proportional to the square of the velocity of a body. Fc α V2 2. Radius of the circle: centripetal force is inversely proportional to the radius of the circle. Fc α 1/ r 3. Mass of the body: centripetal force is directly proportional to the mass of the body Fc α m Combining all above equations Fc α mv2/r Or Fc = mv2/r Centripetal Acceleration: The acceleration produced by centripetal acceleration. According to Newton 2nd law of motion, the direction of centripetal acceleration is along the direction of centripetal force (Fc). 1. Centripetal acceleration is perpendicular to the velocity of the body. 2. Acceleration directed towards the centre of the circle. Fc = mv2/r Fc = mac 2 there fore mac= mv /r ac = V2/ r ac α V2 and ac α 1/r Centrifugal Force: According to Newton 3rd law of motion as a reaction of centripetal force there appears another out ward reaction at the centre of the circle. This force is equal in magnitude to the centripetal force but opposite in direction. This is called centrifugal force. Fr = mv2/r Centrifugal force exits so long as the centripetal force exits. Application of centripetal force. 1. Banking of road. 2. Centrifuge. 3. Washing machine 4. Cream separator. 5. Extraction of juice from sugar cane. GRAVITATION Every object in our universe attracts the other object with certain fore towards its center. This force of attraction is known as GRAVITATIONAL FORCE and the phenomenon is called GRAVITATION. This is gravitational force which is responsible for the uniformity or regularity in our daily astronomical life. The whole system of the universe is in order only due to this force. Due to gravitation, the system of our universe is working uniformly and smoothly. The planets around the earth or around the sun moves in an orderly motion due to gravitation. NEWTON’S LAW OF GRAVITATION In order to explain the gravitational force between two bodies, Newton formulated a fundamental law known after his name i.e. "NEWTON'S LAW OF GRAVITATION" Newton’s law of gravitation states that every object in the universe attracts the other object with a force and : (1) The gravitational force of attraction between two bodies is directly proportional to the product of their masses. 1 x m2 ------- (1) (2) The gravitational force of attraction between two bodies is inversely proportional to the square of the distance between their centers. 2 --------- (2) MATHEMATICAL REPRESENTATION Combining (1) and (2) /d2 F = G m1m2/d2 Where G = universal gravitational constant Value of G: G = 6.67 x 10-11 Nm2/kg2 MASS OF THE EARTH Consider a body of mass ‘m’ placed on the surface of the earth. Let the mass of the earth is ‘Me’ and radius of earth is ‘Re’ . 1m2 Gravitational force of attraction between earth and body is F = G m Me/ Re2 We know that the force of attraction of the earth on a body is equal to weight the weight of body. i.e F=W therefore W = G m Me/ Re2 But W = mg mg = G m Me/ Re2 or g = G Me/Re2 or Me = g x Re2/G From astronomical data: g= 9.8 m/s2 Re = 6.4 x 106 m G = 6.67 x 10-11 N-m2/kg2 Putting these values in the above equation. Me = 9.8 (6.4 x 106)2/6.67 x 10-11 or DENSITY OF EARTH DENSITY OF EARTH We know that the earth is composed of numerous substances such as metals, non-metals, water and other materials. Therefore we cannot determine exact density of the earth .For our convenience, here, we are assuming that the earth is made of homogeneous substance and it has a perfect spherical shape, however which is not true. We know that the density of a substance is given by: density = mass / volume For earth: Density = Me / Ve Let the earth is a sphere of homogeneous composition , therefore its volume is given by: Ve = 4/3(πR3e) Hence the average density of the earth is: Variation of g with altitude The value of ‘g’ is inversely proportional to the distance from the centre of the earth. If distance from the centre of earth increases, the value of g decreases. If value of ‘g’ at height ‘h’ from the earth is gh gh = GM/(R+h)2 gh = gM/(R+h)2 It shows that if distance from the centre of the earth increase from the average radius of the earth , the value of ‘g’ will decreases. This is reason due to which the value of ‘g’ is less on mountain. Orbital velocity of artificial satellite: Due to gravitational force artificial satellite revolve round the earth in different orbits with uniform speed if satellite revolve around the earth at height ‘h’ from the surface of earth. Then centripetal force for satellite is Fc = mv2/R+h ----1 Here m is mass , V is velocity and “R” is radius of earth Gravitation force of earth provides this centripetal force to the satellite F = G m Me/ (R+h)2----------2 Comparing 1 and 2 equations mV2/R+h = GmMe/( R+h)2 V2 = GM(R+h) / ( R+h)2 V= √GM/(R+h) If h the height of satellite is not very large than Vorb= √ GM / R G= 6.67x 10-11 N.m2/KG2 M = 6.4 x 1024 Kg R = 6.4 x 106 m After substituting the value Vorb = 7900ms-1 or 7.9Kms-1