Assignment no. 4 (due: October 8, 2013, Tuesday, 11.59 PM) A look into the centripetal acceleration of Earth
Because of the centripetal forces due to Earth’s rotation, a plumb bob might not hang exactly vertical, along a line orthogonal to the Earth surface and directed to the centre of the Earth, as if it was affected only by the force of gravity. Consider a bob located at a latitude of = 35o North with respect to the Equator. You are asked to determine: a) The centripetal acceleration (aC) due to Earth’s rotation at that latitude b) How much does the bob deviate from a radial line directed towards the centre of Earth (Hint: assume the Earth is spherical with radius R = 6370 km and a rotation period of T = 24 h. Indicate with the angle of deviation from the radial line directed towards the Earth centre). c) How much the bob would deviate at the Equator (E=0) and at the North Pole (N=90o)? d) Assume that aC was much larger than the acceleration of gravity g (e.g. because g was much smaller than what actually is). To which direction the plumb bob would point, North, South, East or West? a) Derive here the symbolic expression for the centripetal acceleration aC as a function of R, T and
Numeric value of aC
b1) List here all of the forces affecting the plumb bob and their direction with respect to a line directed from the location of the bob to the centre of the Earth
The force of gravity is in the same direction as the line from the center of the Earth to the bob. The force created by centripetal acceleration is tangent to the axis of the Earth’s rotation, or at an angle of 35° away from the line directed towards the center of the earth.
b2) Write the equations corresponding to the free-body diagram for the bob For the force of gravity, acting only in the y-direction:
For the x-component of force due to
For the y-component of force due to
Since the plumb bob is not hurtling to the center of the earth, the tension in the string must be equal in magnitude and completely opposite the forces of gravity and centripetal acceleration acting on the plumb bob. In the x-component:
The tension in the y-component is opposite, since my point of reference has the y-components of gravity and centripetal acceleration as positive, I’m going to display the Tension as negative. In the y-component:
b3) Derive here the symbolic expression for as a function of g, R, and T The angle of the bob can be calculated using the arctan of the x and y components of the acceleration acting on the bob without knowing the mass, since the mass is common to both forces. The equation then becomes:
Which can be solved by substituting the equations for Fx and Fy from b2) into the equation above.
Indicate the numeric value of in degrees or radians
Indicate the numeric value of at the North Pole
(N) = 0°
Indicate the numeric value of at the Equator
(E) = 0°
Please justify your answer
At the North Pole there is no centripetal acceleration; objects at the pole do not rotate at around anything so the radius is zero. Since there is no centripetal acceleration the only force acting on the plumb bob in this situation is the force of gravity At the equator centripetal acceleration is perfectly in line with the acceleration from gravity, the plumb bob points straight to the center of the earth with an acceleration slightly greater than 9.8 m/s2.
Direction of the plumb bob if it was aC
Please join StudyMode to read the full document