Question

You hang two pendulums next to one another. They both are 3 meters long.

One of them has a 1 kg mass at the bottom, the other has a 2 kg mass at

the bottom. You pull the lighter ball back so that it forms a 40◦ angle with

the vertical, and then let it go. It collides elastically with the other pendulum.

What is the maximum angle the heavier pendulum creates with the vertical?

Answer 1

Line AB is the minimum gravitational potential energy line on which we define the potential of any mass to be zero .

Along the line AB :

Just Before collision : Only 1kg ball is moving with velocity v1

Just after collision : 3 kg ball is moving with velocity v2 and 1 kg with v1 ( but in opposite direction )

-444444 360540° 3 coso 3m to 3m 12 from the figure above, we can see 3= h + 3 Los 400 h = 3 - 3 cos 40° = 3-2.298 h = 0.702 m Let the velocity of 1 kg mass just before the collision is v e So, when release 1 kg from height h = 0.702 m it starts coming down and just before collision all its potential energy converts into Kinetic energy. So migh, I m, vt 1x 10x 0.702 = 1 X 1 XV, 2 2 ~ = √2X1080:702 = 3.746 m/s

Let, just after collision, the velocity of 3 kg mass be v₂ So, Conservating linear momentum along the line AB Initial momentum = final momentum along along AB - AB 1x 3.746 = 1X (63) + 3xV2 - 0 Here, Uz ie the velocity of 1 kg just after the collision. c-ve) sign represents that direction of V₂ is just opposite to that of V2:

coefficien for the of where, e case of Elastic collision, Restitution , e=1 e is defined as velocity of seperation velocity of approach = V₂-(-V3) = - v.-0 V2 + V3 ③ v = ₂ + V3 3.746 = 2 + V3 = - V3 = V₂ -3.746 substituting this value in equation 3.746= V₂ - 3.746 + 3 V2 4 v₂ = 2 (3.746) V = 1.873 mis

Maximum height ₂ 3 kg ball, when collision has been is achieved by the all its [(KE) just after converted into Potential Energy so £ ma ve 2 = maghz h₂ = ₂² (1.873)² 2o h₂ = 0.175 m from the figure on first page , we can clearly see that 3 =h, t 3 cos o where, o is the maximum angle made by ball with vertical. so 3 coso = 3-0.175 = 2.825 coso = 2.825 = 0.9416 o = cos (0.9416) To = 19.6170