Types of impact; collar on vertical rod

Types of impact

Impact is a collision of two bodies in a very short time, producing large forces. The line through the contact point, normal to the surfaces, is the line of impact.

• Central impact: the centres of mass of both bodies lie on the line of impact. It is direct if velocities are along this line and oblique if they are inclined to it.
• Eccentric impact: the line of impact does not pass through the centre of mass of one or both bodies.
• By the coefficient of restitution $e$: perfectly elastic ($e=1$, no energy loss), perfectly plastic ($e=0$, bodies stick together) and partly elastic ($0<e<1$).

Collar on the vertical rod

Assumptions (figure not shown): the spring ($k=100$ N/m, free length 1 m) is attached to a wall point level with $s=0$ and 1.5 m horizontally from the rod. The collar moves down the rod. Only the weight and the spring do work.

 wall   1.5 m   rod
  o-------------A  s = 0
   \            |
    \  spring   |  s
     \          |
      \_________B

Spring stretches (using $\ell=\sqrt{1.5^{2}+s^{2}}$):

• at $s=0$: $\ell=1.5$ m, stretch $=0.5$ m
• at $s$: stretch $=\ell-1$

Work-energy from $s=0$ to the momentary stop ($v=0$):

$$\tfrac12(4)(3)^{2}+4(9.81)s-\tfrac12(100)\left(\sqrt{1.5^{2}+s^{2}}-1\right)^{2}+\tfrac12(100)(0.5)^{2}=0$$ $$18+39.24s+12.5=50\left(\sqrt{2.25+s^{2}}-1\right)^{2}$$

Solving numerically gives $s=1.955$ m. Check: $\ell=\sqrt{2.25+1.955^{2}}=2.464$ m, so $50(1.464)^{2}=107.2$ J and the left side is $30.5+76.7=107.2$ J. The check holds.

Answer: the collar stops momentarily after travelling $s\approx1.96$ m.