Time dilation on Jupiter due to revolution

TIME DILATION ON JUPITER’S SURFACE DUE TO REVOLUTION

INTRODUCTION

Jupiter is the 5^th planet in our Solar System. It’s 778 million km [486 million miles] from the Sun. It completes one revolution around the Sun in approximately 11.86 Earth years. Consider the planet Jupiter which revolves at a certain velocity and another analogous Jupiter which doesn’t revolve. According to the Special Theory of Relativity, a clock on revolving Jupiter would run slower than that on a non-revolving Jupiter. We’ll find the time gained by revolving Jupiter relative to the stationary one.

ASSUMPTIONS

• The revolution path is exactly circular.

• Jupiter is a perfect homogeneous sphere.

• The effect of Gravitational time dilation is negligible.

CALCULATION

The Angular velocity is given by,

ω = 2π/T [rad/s]

T – Revolution time period [T ≈ 11.86 years = 374,016,960 s]

ω = 2*3.14/374016960

ω = 1.6799*10^-8 rad/s

The tangential velocity is given by,

v = R*ω [m/s]

R – Average distance between Sun and Jupiter [R = 778 million km]

v = 778*10⁶*1000*1.6799*10^-8

v = 13069.622 m/s

According to the Special Theory of Relativity, the time dilation equation is,

t' = t/γ [s]

t’ – Actual time or Moving observer’s time. [s]

t - Proper time or Stationary observer’s time. [s]

γ – Relativistic gamma factor, γ = 1/√ [1-(v/c) ²]

c - Velocity of light [c = 3*10⁸ m/s]

t' = t*√ [1-(v/c) ²]

t' = t*√ [1-1.8979*10^-9]

t' = t*√ [0.9999999981021]

t' = t* 0.99999999905105

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on revolving Jupiter relative to the stationary one. We’ll consider 5 different t’ values and calculate t value. The larger the t’ the more is the difference between t and t’. Thus revolving Jupiter will gain time over the stationary one. In fact in one year it gains 354 milliseconds just due to revolution and 26 milliseconds due to rotation. The time gained by revolution is more than rotation because Jupiter revolves at 13 km/s as compared to rotation which is 12.2 km/s.

Time	t’ [Stationary Jupiter] (s)	t [Revolving Jupiter] (s)	Difference (s)
1 minute	60	59.999999943063	0.000000056937
1 hour	3600	3599.99999658378	0.00000341622
1 day	86400	86399.9999180107	0.0000819893
1 month	2592000	2591999.99754032	0.00245968
1 year	31536000	31535999.9700739	0.0299261
11.86 years	374016960	374016959.645077	0.354923

Time dilation on Jupiter's surface due to rotation

TIME DILATION ON JUPITER’S SURFACE DUE TO ROTATION

INTRODUCTION

Jupiter is the 5^th planet in our Solar System. It completes one rotation about its axis in 9 hours 55 minutes and 30 second. It is also the fastest rotating planet in our Solar System. Consider the planet Jupiter which rotates at a certain velocity and another analogous Jupiter which doesn’t rotate. According to the Special Theory of Relativity, a clock on rotating Jupiter would run slower than that on a non-rotating Jupiter. We’ll find the time gained by rotating Jupiter relative to the stationary one.

ASSUMPTIONS

Jupiter is not revolving around the Sun.

Jupiter is a perfect homogeneous sphere.

The effect of Gravitational time dilation is negligible.

CALCULATION

The Angular velocity is given by,

ω = 2π/T [rad/s]

T – Rotational time period [T = 9 hours 55 minutes 30 second = 35730 s]

ω = 2*3.14/35730

ω = 1.7585*10^-4 rad/s

The angular velocity is same at all points on Jupiter since it doesn’t exhibit differential rotation. But the tangential velocity on surface varies with the distance from the center.

The tangential velocity is given by,

v = R*ω [m/s]

R – Average radius of Jupiter [R = 69911 km]

v = 69911*1000*1.7585*10^-4

v = 12,293.8493 m/s

According to the Special Theory of Relativity, the time dilation equation is,

t' = t/γ [s]

t’ – Actual time or Moving observer’s time. [s]

t - Proper time or Stationary observer’s time. [s]

γ – Relativistic gamma factor, γ = 1/√ [1-(v/c) ²]

c - Velocity of light [c = 3*10⁸ m/s]

t' = t*√ [1-(v/c) ²]

t' = t*√ [1-1.6793*10^-9]

t' = t*√ [0.9999999983207]

t' = t* 0.99999999916035

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on rotating Jupiter relative to the non-rotating one. We’ll consider 5 different t’ values and calculate t value. The larger the t’ the more is the difference between t and t’. Thus rotating Jupiter will gain time over the non-rotating one. In fact it’ll gain 26 milliseconds in one year.

Time	t’ [Non-rotating Jupiter] (s)	t [Rotating Jupiter] (s)	Difference (s)
1 minute	60	59.999999949621	0.000000050379
1 hour	3600	3599.99999697726	0.00000302274
1 day	86400	86399.9999274542	0.0000725458
1 month	2592000	2591999.99782363	0.00217637
1 year	31536000	31535999.9735208	0.0264792

Mesh Weighing scale

MESH WEIGHING SCALE

INTRODUCTION 

Designed and developed a new weighing machine known as Mesh weighing scale. It consists of a plastic sheet in tension. When mass is placed on the sheet, the sheet distorts. The distortion is measured by a suitable scale; this reading is proportional to the object’s mass.

APPARATUS:

Mesh made of plastic, 4 poles, base, linear weights and ruler.

APPARATUS DESCRIPTION:

Mesh: It’s a plastic sheet 5’x5’.

Poles: They’re 4’ long rods which hold the plastic sheet. These rods are glued to the base.

Conical tops: They’re glued on top of the poles so as to hold the sheet. The sheet is never glued due to the need of replacing them frequently.

Ruler: It’s a linear rod which is calibrated to measure the distortion in the sheet.

Linear Weights: Any weight that’s roughly linear.

PROCEDURE:

The apparatus is arranged as shown in the picture. The weight needs to be placed perfectly linear near the ruler.

MEASUREMENT:

The weight bends the plastic sheet which in turn moves and coincides with the ruler. We can make an approximate measurement using this method. Any linear weights can be measured; they need not be perfectly linear like rectangular rod. We can measure mass of pencil, pen, toothpaste tube, etc. Note that the plastic sheet must be replaced after certain measurements as it can give wrong readings since plastic loses its small elastic strength to return back to its initial position. 

CONCLUSION:

Thus we can measure linear weights easily using this equipment without using electricity!!!