Pendulum's and special relativity

[k4wedi]

1) A mathematical pendulum , length L= 0.61 m oscillates in a uniformly accelerating vehicle. The acceleration is horizontal and equal to 3.2 m/ s^2 The period of oscillations is?

I don't know where to put in the acceleration when it is horizontal. I thought that if I just added the two vectors mg and ma I would get the answer but that turned out to be the same as if I was moving up vertically..

2) At what speed v should an object move with respect to a system S in order that its density measured from S is n=1.088 times greater then its proper density ( i.e. density measured in a system in which object is at rest ). Express your result as a ratio v/c .

I got it so that: 1.088(RestDensity) = (RestDensity) / [ 1 - (v/c)^2 ]

After working it out I got my answer to be 0.284... but the "right" answer is 0.878.. that doesn't make any sense to me since you'd have to move at almost 90% the speed of light just to increase 1.088 times your rest density??

3) Calculate the speed at which the kinetic energy of a particle is equal to its rest energy. Express your result as v/c

This.. I got it to be 0.84 and the "right" answer is 0.87..

I used E^2 = p^2 * c^2 + m^2*c^4; where m is the rest mass and p is the momentum
equated p^2*c^2 to m^2*c^4 and solved for v by subbing p = mv where the m in this case is the relativisic mass. Is that just plain wrong?

Much thanks to anyone who can help, I've been at these problems for a while.. :<

 

[OlderDan]

For #1 figure out where the equilibrium position will be (at an angle to the vertical) and find the "effective g" which is related to the tension in the string as viewed by an observer in the vehicle. As viewed from outside the vehicle, the horizontal component of the tension is responsible for the acceleration.

What you did for #2 seems right. I assume you are to use relativistic mass and the volume decreases because of length contraction, so your factor of gamma² looks right.

For #3 it appears you may have two problems. It looks like you set the square of the total energy equal to twice the square of the rest energy. I think what you need is (E - m_o*c²) = kinetic energy = 2*m_o*c² or E = 3*m_o*c². Unfortunately, that does not give you the "right" anser either. The supposed right answer is setting the total energy (not the kinetic energy) to twice the rest energy. That is not correct.

 

[kyle8921]

I would like to address the issue of pendulums and special relativity. First, let's start by clarifying that special relativity deals with the behavior of objects moving at constant speeds in a straight line, while pendulums involve oscillations and circular motion. Therefore, the two concepts may not seem directly related, but they can both be applied in certain scenarios.

Regarding the first problem, the fact that the acceleration is horizontal does not affect the period of oscillations. This is because the period of a pendulum is determined by its length and the acceleration due to gravity, which is always directed downwards. In this case, the acceleration due to gravity is still acting on the pendulum, and the horizontal acceleration only adds to it, resulting in a slightly larger total acceleration. The period of oscillation will not change significantly due to this added acceleration.

Moving on to the second problem, it involves the concept of time dilation in special relativity. The equation you have used is correct, but it seems that you have made a mistake in your calculations. The correct answer should be 0.878, which means that the object would need to move at 87.8% of the speed of light to have a density 1.088 times greater than its rest density. This may seem like a high speed, but keep in mind that objects moving at high speeds experience time dilation, meaning that time moves slower for them. This is a well-established phenomenon in special relativity and has been verified by numerous experiments.

Finally, in the third problem, you have correctly used the equation for relativistic energy. However, the correct answer should be 0.87, not 0.84. This means that the speed at which the kinetic energy of an object is equal to its rest energy is 87% of the speed of light. Again, this may seem like a high speed, but it is in line with the predictions of special relativity.

In conclusion, special relativity is a well-tested and validated theory that has been used to explain many physical phenomena, including the behavior of objects at high speeds and the effects of gravity. While it may seem counterintuitive at times, the equations and principles of special relativity have been repeatedly verified by experiments and are an essential part of modern physics.