Gravity and Angular acceleration

[sreya]

Homework Statement

There is strong evidence that Europa, a satellite of Jupiter, has a liquid ocean beneath its icy surface. Many scientists think we should land a vehicle there to search for life. Before launching it, we would want to test such a lander under the gravity conditions at the surface of Europa. One way to do this is to put the lander at the end of a rotating arm in an orbiting Earth satellite.

If the arm is 5.25m long and pivots about one end, at what angular speed (in rpm) should it spin so that the acceleration of the lander is the same as the acceleration due to gravity at the surface of Europa? The mass of Europa is 4.8E22kg and its diameter is 3138 km.

$\omega$ =_____rpm

Homework Equations

$v=\omega r$

$\frac{GMm}{R^2}=a$

$T = \frac{2\pi}{\omega}$

The Attempt at a Solution

$\frac{GMm}{R^2}=a$

$\frac{mv^2}{R}=ma$

$\frac{\omega^2R}{a}$

$\omega=\sqrt{\frac{a}{R}}$

$\omega=\sqrt{\frac{GM}{R_{europa}^2*R_{sat}}}$

$\frac{60\omega}{2\pi}$ = x rpm

Edit: Figured out the problem. The diameter of Europa is given in Km, you have to convert it to meters. Stupid Mastering Physics...

Apparently that's not right though??

 

[Simon Bridge]

$\frac{GMm}{R^2}=a$

... this is not correct: dimensions don't match.

Your reasoning is unclear - you seem to want to put the centripetal acceleration of the station centrifuge equal to the acceleration due to gravity at the surface of Europa.

Try writing centripetal acceleration in terms of angular velocity.

 

[sreya]

Sorry that should be
$\frac{GMm}{R^2}=g_{europa}$

Which "technically" is still acceleration but that wasn't clear

 

[Simon Bridge]

Still not correct.
Dimension still don't match.

Does the acceleration of gravity depend on the mass of the object falling?
Hint: leaning tower of Pisa.

What about writing centripetal acceleration in terms of angular velocity?

 

[HalfLostAndSearching]

I would first like to commend the student for their attempt at a solution and for recognizing their mistake in converting the diameter of Europa to meters. However, it seems that the student's final answer is still incorrect.

To find the angular speed (ω) in rpm, we first need to calculate the acceleration due to gravity (a) at the surface of Europa using the given mass and diameter. We can use the formula a=GM/R^2, where G is the universal gravitational constant, M is the mass of Europa, and R is the radius of Europa (which can be calculated from the given diameter).

Once we have the value for a, we can use the formula ω=√(a/R) to find the angular speed in rad/s. We can then convert this to rpm by multiplying by 60/2π.

Therefore, the correct solution is:

a = (6.674×10^-11 N*m^2/kg^2)(4.8×10^22 kg)/(1.569×10^6 m)^2 = 1.315 m/s^2

ω = √(1.315 m/s^2)/(5.25 m) = 0.309 rad/s

ω = (0.309 rad/s)(60/2π) = 9.32 rpm

So, the lander should spin at 9.32 rpm to experience the same acceleration as the surface of Europa. This is a crucial step in testing the lander before launching it to Europa, as it ensures that the lander will be able to withstand the gravity conditions on the surface.