1) The potential energy of a particle is given in Joules as a function of its po

Question

1) The potential energy of a particle is given in Joules as a function of its position x: U(x) = 5(x 2)^2 + 2x, where x is in meters. If the total mechanical energy of the particle is 10 J find its kinetic energy when the acceleration of the particle is zero

2) A 10 kg table stands on a floor on four legs. If each of the legs has a mass of 1 kg and a length of 1 m how high above the floor (in meters) is its center of mass?

3) A 4 kg package gliding on a horizontal frictionless surface explodes into two 2 kg fragments. One of them is moving at 3 m/s due north and the other one is moving at 5 m/s in a direction 30 degrees north of east. What is the original speed of the package?

Explanation / Answer

1.) U(x) = 5(x 2)2 + 2x

Total mechanical energy = 10 J

dU /dx = Force

when acceeration is Zero i.e force is zero

dU /dx = Force = 0

U(x) = 5(x 2)2 + 2x

dU /dx = 10(x 2) + 2 = 0

x = -0.2 + 2 = 1.8 m

Total Energy = 10

So 10 = 5(x 2)2 + 2x + KE at x = 1.8 m

10 = 5(1.8 2)2 + 2*1.8 + KE

or KE = 6.2 joule

Hence kinetic energy when the acceleration of the particle is zero = 6.2 joule

2) Mass of table = 10 kg

four legs each of the legs has a mass of 1 kg and a length of 1 m

Center of mass = (4*0.5 + 6*1 ) /10 = 0.8 m from ground

3) V1 = 3 m/s North V2 = 5 m/s 30 degrees north of east

Momentum conservation East direction

4 *Vx + 2* 5 *cos(30) = 0

Vx = -2.17 m/s

North direction 4*Vy + 3*2 + 5Sin(30) = 0

Vy = -2.125 m/s

original speed = ( Vx2 + Vy2 )1/2 = 3.03 m/s

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