As we saw in
Applications of Newton’s Laws , objects moving at constant speed in a circle have a centripetal acceleration directed toward the center of the circle, which means that there must be a net force directed toward the center of that circle. Since all objects on the surface of Earth move through a circle every 24 hours, there must be a net centripetal force on each object directed toward the center of that circle.
Let’s first consider an object of mass
m located at the equator, suspended from a scale (
[link] ). The scale exerts an upward force
${\overrightarrow{F}}_{\text{s}}$ away from Earth’s center. This is the reading on the scale, and hence it is the
apparent weight of the object. The weight (
mg ) points toward Earth’s center. If Earth were not rotating, the acceleration would be zero and, consequently, the net force would be zero, resulting in
${F}_{\text{s}}=mg$ . This would be the true reading of the weight.
With rotation, the sum of these forces must provide the centripetal acceleration,
${a}_{\text{c}}$ . Using Newton’s second law, we have
Note that
${a}_{\text{c}}$ points in the same direction as the weight; hence, it is negative. The tangential speed
v is the speed at the equator and
r is
${R}_{\text{E}}$ . We can calculate the speed simply by noting that objects on the equator travel the circumference of Earth in 24 hours. Instead, let’s use the alternative expression for
${a}_{\text{c}}$ from
Motion in Two and Three Dimensions . Recall that the tangential speed is related to the angular speed
$\left(\omega \right)$ by
$v=r\omega $ . Hence, we have
${a}_{c}=\text{\u2212}r{\omega}^{2}$ . By rearranging
[link] and substituting
$r={R}_{\text{E}}$ , the apparent weight at the equator is
Substituting for the values or
${R}_{\text{E}}$ and
$\omega $ , we have
${R}_{\text{E}}{\omega}^{2}=0.0337\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$ . This is only 0.34% of the value of gravity, so it is clearly a small correction.
Zero apparent weight
How fast would Earth need to spin for those at the equator to have zero apparent weight? How long would the length of the day be?
Strategy
Using
[link] , we can set the apparent weight (
${F}_{\text{s}}$ ) to zero and determine the centripetal acceleration required. From that, we can find the speed at the equator. The length of day is the time required for one complete rotation.
Solution
From
[link] , we have
$\sum F={F}_{\text{s}}}-mg=m{a}_{\text{c}$ , so setting
${F}_{\text{s}}=0$ , we get
$g={a}_{\text{c}}$ . Using the expression for
${a}_{\text{c}}$ , substituting for Earth’s radius and the standard value of gravity, we get
We will see later in this chapter that this speed and length of day would also be the orbital speed and period of a satellite in orbit at Earth’s surface. While such an orbit would not be possible near Earth’s surface due to air resistance, it certainly is possible only a few hundred miles above Earth.
vector has both magnitude & direction but scalar has only magnitude
Manash
my marunong ba dto mag prove ng geometry
ron
how do I find resultant of four forces at a point
Inusah
use the socatoa rule
kingsley
draw force diagram, then work out the direction of force.
Rongfang
In a closed system of forces... Summation of forces in any direction or plane is zero... Resolve if there is a need to then add forces in a particular plane or direction.. Say the x direction... Equate it tk zero
a system of mechanics based of Newton laws motion this is easy difenation of mean...
Arzoodan
what is the meaning of single term,mechanics?
jyotirmayee
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
Lalita
SO ASK
What is Newtonian mechanics in physics?
Newtonian physics, also calledNewtonian or classical mechanics, is the description of mechanical events—those that involve forces acting on matter—using the laws of motion and gravitation formulated in the late seventeenth century by English physicist
Suleiman
can any one send me the best reference book for physics?
Prema
concept of physics by HC verma, Fundamentals of Physics, university of physics
Bhaskar
tq u.
Prema
these are the best physics books one can fond both theory and applications.
Bhaskar
can any one suggest best book for maths with lot of Tricks?
Vivek
what is the water height in barometer?
SUNEELL
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
LOVE
water is 13.5 times dense than the Mercury
LOVE
plz tell me frnds the best reference book for physics along with the names of authors.
Prema
i recomended the reference book for physics from library University of Dublin or library Trinity college
Arzoodan
A little help here...
.
1. Newton's laws of Motion, are they applicable to motions of all speeds?
2.state the speeds which are applicable to Newtons laws of Motion
Derek
mechanics which follows Newtons law
Manash
The definition of axial and polar vector .
Arpita
polar vector which have a starting point or pt. of applications is,force,displacement
jyotirmayee
axial vector represent rotational effect and act along the axis of rotation b
It states that, " If two vectors are represented in magnitude and direction by the two sides of a triangle, then their resultant is represented in magnitude and direction by the third side of the triangl " .