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Chapter 6: Q6.1-4PE (page 221)

(a) What is the period of rotation of Earth in seconds?

(b) What is the angular velocity of Earth?

(c) Given that Earth has a radius of6.4脳106mat its equator, what is the linear velocity at Earth鈥檚 surface?

Short Answer

Expert verified

a) The period of rotation of the earth is 86400 s.

b) The angular velocity of the earth is 7.291159脳10-5rad/s.

c) The linear velocity at the earth鈥檚 surface is 470鈥�m/s.

Step by step solution

01

Definition of linear velocity

The actual value of the period of rotation of the earth is 23 hours, 56 minutes, and 4 seconds.

This happens because a solar day is longer than a sidereal day. While the earth rotates, it also moves around the sun with an interval of one day.

02

Calculation of the period of the rotation of the earth

The period of rotation of the earth can be calculated as:

=1脳24脳60脳60鈥剆=86400鈥剆

So, the period of the rotation of the earth is 86400 s.

03

Calculation of the angular velocity of the earth

The angular velocity can be calculated as:

=2蟺86400=7.275脳10鈭�5鈥剅补诲/蝉

The earth rotates at a moderate angular velocity of 7.275脳10鈭�5鈥剅补诲/蝉.

04

Calculation of the linear velocity of the earth

Earth鈥檚 radius=6.4脳106鈥�m.

Thus,the linear speed of any point on the earth鈥檚 surface at the equator due to the earth鈥檚 rotation is

V=r蝇=6.4脳106脳7.3脳10鈭�5=470鈥�m/s

Therefore, the linear velocity at the earth鈥檚 surface is 470鈥�m/s.

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Most popular questions from this chapter

When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the northern hemisphere. (Note that this is a small effect and in most toilets the rotation is caused by directional water jets.) Would the direction of rotation reverse if water were forced up the drain?

Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components鈥攆riction parallel to the road (this must supply the centripetal force), and the vertical normal force (which must equal the system鈥檚 weight).

(a) Show that\(\theta \)(as defined in the figure) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway鈥攖hat is,\(\theta = {\tan ^{ - 1}}\,{v^2}/rg\)

(b) Calculate \(\theta \) for a \(12.0{\rm{ m}}/{\rm{s}}\) turn of radius \(30.0{\rm{ m}}\) (as in a race).

Figure 6.36 A bicyclist negotiating a turn on level ground must lean at the correct angle鈥攖he ability to do this becomes instinctive. The force of the ground on the wheel needs to be on a line through the center of gravity. The net external force on the system is the centripetal force. The vertical component of the force on the wheel cancels the weight of the system while its horizontal component must supply the centripetal force. This process produces a relationship among the angle \(\theta \), the speed \(v\), and the radius of curvature \(r\) of the turn similar to that for the ideal banking of roadways.

An automobile with 0.260 m radius tires travels80,000km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?

Space debris left from old satellites and their launchers is becoming a hazard to other satellites

(a) Calculate the speed of a satellite in an orbit900 km above Earth鈥檚 surface.

(b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite鈥檚 orbit at an angle of90掳relative to Earth. What is the velocity of the rivet relative to the satellite just before striking it?

(c) Given the rivet is 3.00 mmin size, how long will its collision with the satellite last?

(d) If its mass is0.500 g, what is the average force it exerts on the satellite? (e) How much energy in joules is generated by the collision? (The satellite鈥檚 velocity does not change appreciably, because its mass is much greater than the rivet鈥檚.)

If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed to take a \(100{\rm{ m}}\) radius curve banked at \(15.0^\circ \). (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at \(20.0{\rm{ km}}/{\rm{h}}\)?
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