5.1 Centripetal Force:

• The velocity of a body changes direction when a force acting on the body is perpendicular to the velocity.
• As the trajectory or path bends, the applied force is pointing toward the center of each segment of the curved path. Such force is called centripetal force Fc. The acceleration associated with the centripetal force is the centripetal accerleration ac. They are related by Fc = m ac.
• The magnitude and direction of the Centripetal Acceleration can be derived as shown in the diagram below.
• The resulting Centripetal Force can be written as
  
  ac = [mV**2 / r] (-r/r)  
  
  where r/r is a unit vector of magnitude 1.0 in the radial direction.

5.2  Centripetal Force:  Fc = m ac Ex: Compute the Earth's centripetal acceleration toward the Sun. The time it takes to go around the Sun once is 365 days. The radius on average is R = 1.5x10**8 km. The average orbital speed of the Earth is V = 2pR/365x24x60x60 s = 30 km/s ac = V**2/R = 900 (km**2/s**2) / 1.5x10**8 km = 600x10**-8 km/s**2 = 6x10**-3 m/s**2 pointing to the Sun. The centripetal force between the Sun and the Earth is Fc = ma = 6x10**24 kg x6x10**-3 m/s**2 = 36x10**21 N

 5.3 The Law of Universal Gravitation:

Newton's law of universal gravitation states that the gravitational force attracting two bodies of mass m and M separated by a distance "r" is given by

          Fg = G m M / r**2 [N]

where G = proportionality constant = 6.67 x 10**-11 [N m**2/kg**2] is an experimentally determined constant.

The universal gravitation force is proportional to the product of the mass of the two bodies and inversely proportional to the distance squared separating the two bodies.

The Earth stays in its orbit due to the centripetal force exerted on the Earth by the Sun. The example in Section 5.2 showed that the centripetal force

           Fc = 36x10**21 (-r/r)  in N.

This centripetal force can only be provided by the gravitational pulling on the earth by the Sun,

   FG = G m M/r**2 = 6.67x10**-11 x 6x10**24 kg x 2x10**30 kg/(1.5x10**11 m)**2
        = 35.6x10**(-11+24+30-22) N = 35.6x10**21 N

This confirms that Fc = FG

5.4 Earth's Gravitational Acceleration

    At the sea level, r = 1Re

           mg(r=1Re) = G mM/Re**2    where M= Earth's mass, Re = Earth's Radius

           g(1Re) = GM/Re**2          
                   = 6.7x10**-11 x 6x10**24 kg/(6.4x10**6 m)**2
                   = 0.98x10**(-11+24-12) m/s**2 = 9.8 m/s**2

    At 2 earth radii above the sea level,

             g(r=3Re) = G M/(3Re)**2 = g(1Re)/9 = 1.1 m/s**2

HW Chapter 5: P9, P23, P25, P33