Answer to Question #281394 in Mechanics | Relativity for Voldemort

Question

1. A pinball machine launch ramp consisting of a spring and a 30 degree ramp of length 𝐿 as shown in Fig. 1.

(a) (3 marks) If the spring is compressed a distance 𝑥 from its equilibrium position and is then released at 𝑡 = 0, the pinball (a sphere of mass 𝑚 and radius 𝑟) reaches the top of the ramp at 𝑡 = 𝑇. Derive the expression for the spring constant 𝑘 in terms of 𝑚, 𝑔, 𝑥, and 𝐿. [Assume that the friction is sufficient, and the ball begins rolling without slipping immediately after launch.]

(b) (2 marks) What is the potential energy of the ball when it is at the midpoint of the ramp?

(c) (3 marks) Derive the expression of the speed of the ball immediately after being launched in terms of 𝑔 and 𝐿.

Answer 1

Given quantities:

$\alpha = 30^o \space\space L, x, t =0, m, r, k$

1) From figure

$H = Lsin\alpha = \large\frac{L}{2}$

by conservation of energy

$\frac{1}{2}kx^2=mgH$

$\frac{1}{2}kx^2=mg\frac{L}{2} \to x = \large\sqrt{\frac{mgL}{k}}$ $W_p = \frac{kx^2}{2}$

2) $\large\frac{mv^2}{2}=\frac{mgL}{2} \to v = \sqrt{gL}$