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.