Olympiad problems

2010 F = Ma, 16

Tags: 2010 , problem 16

Following the previous set up, find the speed $v$ of the small block after it leaves the slope. (A) $v=v_\text{0}$ (B) $v=\frac{m}{m+M}v_\text{0}$ (C) $v=\frac{M}{m+M}v_\text{0}$ (D) $v=\frac{M-m}{m}v_\text{0}$ (E) $v=\frac{M-m}{m+M}v_\text{0}$

2009 F = Ma, 16

Tags: 2009 , problem 16

Two identical objects of mass $m$ are placed at either end of a spring of spring constant $k$ and the whole system is placed on a horizontal frictionless surface. At what angular frequency $\omega$ does the system oscillate? (A) $\sqrt{k/m}$ (B) $\sqrt{2k/m}$ (C) $\sqrt{k/2m}$ (D) $2\sqrt{k/m}$ (E) $\sqrt{k/m}/2$

2008 F = Ma, 16

Tags: 2008 , problem 16

A [i]massless[/i] spring with spring constant $k$ is vertically mounted so that bottom end is firmly attached to the ground, and the top end free. A ball with mass $m$ falls vertically down on the top end of the spring, becoming attached, so that the ball oscillates vertically on the spring. What equation describes the acceleration a of the ball when it is at a height $y$ above the [i]original[/i] position of the top end of the spring? Let down be negative, and neglect air resistance; $g$ is the magnitude of the acceleration of free fall. (a) $a=mv^2/y+g$ (b) $a=mv^2/k-g$ (c) $a=(k/m)y-g$ (d) $a=-(k/m)y+g$ (e) $a=-(k/m)y-g$

2011 F = Ma, 16

Tags: 2011 , problem 16

What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping? (A) $Mg$ (B) $\mu_k Mg$ (C) $\mu_k Mg/\mu_s$ (D) $(\mu_s + \mu_k)Mg$ (E) $Mg/\mu_s$