Tangential velocity of the earth

[RTW69]

Homework Statement

Determine the speed with which the Earth would have to turn to rotate on its axis so that a person on the equator would weigh 3/4 as much

Homework Equations

V_T=r*ω ; V_i=469 m/s is tangential velocity of earth

ƩF=M*a_c=m*V_t^2/r

The Attempt at a Solution

The positive direction is toward the center of the earth.

From ƩF=m*a_c

Initial: m*g-N_i=m*V_i^2/r

Final: 3/4m*g-N_f=m*V_f^2/r

Since m*g is the same for initial and final state I assume that N_i=N_f

Therefore:3/4m*g-[mg-m*V_i^2/r]=m*V_f^2/r or

V_f=Sqrt(V_i^2-r*g/4)

I have a sign error. I end up taking the square root of a negative number but the physics looks OK. Suggestions?

 

[haruspex]

Nf is to be 3/4 of what? And what equation tells you what it will actually be? (Your 'Final' equation is completely wrong.)

 

[RTW69]

OK, Final:m*g-3/4*N=m*V_f^2/r

 

[haruspex]

Looks right, if N is what you wrote as Ni previously,

 

[Nickie]

I appreciate your attempt at finding a solution to this problem. However, I would like to clarify a few things. The tangential velocity of the Earth at the equator is not a constant value, as it varies with latitude due to the Earth's oblate shape. Additionally, the weight of an object is influenced by the gravitational force, which is dependent on the mass and distance of the object from the center of the Earth. Therefore, the weight of an object on the equator would not be affected solely by the Earth's rotation.

That being said, if we assume that the Earth's rotation is the only factor affecting weight, your approach using the equations of motion is correct. However, there may be a mistake in your calculations, which could be causing the sign error. I would suggest double-checking your calculations and units to ensure accuracy. Additionally, it may be helpful to consider the Earth's rotational speed at the equator, which is approximately 1670 km/h, and see how it compares to the value you have calculated.

In conclusion, while your approach is scientifically sound, it is important to consider all factors and double-check calculations to ensure accuracy. Keep up the good work!