Newton's Cradle W/ Motion Study (Elastic Collision)

grabcad

Here's an example of a physics problem involving a Newton's Cradle:A “Newton’s Cradle” is a toy consisting of 5 steel balls suspended by strings from a metal frame as shown in the diagram. Each of the balls has diameter R and mass m, and the strings are each of length L. The 1st ball (i.e., the one on the far left) is pulled back and released from rest so that the string supporting it is under tension throughout the ball’s motion; the other 4 balls are all holding still during this time. You may assume R << L.a) If the magnitude of the momentum of the 1st ball is p_1 immediately before it strikes the 2nd ball, then what was the initial angle θ between the string and the vertical as viewed in the diagram? Express your answer in terms of any subset of p_1, m, R, L, and any relevant physical constants.The trick to this is knowing how to simplify the variables in the conservation of energy equation. If we choose our two points as 1) before the first ball is dropped and 2) immediately before it collides with the second ball, then here is the conservation of energy equation:mgy_1 = 1/2mv_1^2where y_1 is the difference in heights between the first ball and the second ball and v_1 is the velocity of the ball immediately before it collides with the second ball.After doing a lot of trigonometry and geometrical analysis (triangles were key here), we find that the first ball is a vertical distance L-Lcosθ above the second ball. Because momentum is purely one-dimensional before the first ball collides with the second:p_1 = mv_1, which implies v_1 = p_1/mAfter plugging in these values into the conservation of energy equation and doing a lot of algebra, we find that the initial angle is:θ = arccos(1-p_1^2/(2m^2gL))b) What is the tension in the string supporting the 1st ball immediately before it strikes the 2nd ball? Express your answer in terms of any subset of p1, m, R, L, and any relevant physical constants.This part is a lot less dense. In order to find the tension, set up a force equation immediately before the first ball collides with the second and define the radial direction to be parallel to the tension force and the tangential direction to be perpendicular to the tension force:T-mg = ma_rwhere T is the tension force and a_r is the radial acceleration of the first ball.Because a_r is radial, we can say:a_r = v^2/LUsing the relation v = p_1/m, we can simplify the force equation to:T = p_1^2/mL+mgNOTE: Can anybody explain why they collide so inconsistently after the first few collisions? After examining it myself, I found that the fifth ball (on the right) indents itself after the second collision, which then impacts all the other collisions. Any help would be appreciated.