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phy131studiof15:lectures:m2review1 [2015/10/22 12:04] mdawber [Ballistic Pendulum] |
phy131studiof15:lectures:m2review1 [2015/10/23 09:16] mdawber [Conditions for Equilibrium] |
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Where the torques can be calculated around **any** axis. Some axes are more sensible than others, a good idea is to choose an axis at place where a force is acting (as the torque due to that force around that axis will be zero.) | Where the torques can be calculated around **any** axis. Some axes are more sensible than others, a good idea is to choose an axis at place where a force is acting (as the torque due to that force around that axis will be zero.) | ||

+ | ===== Hanging sign solution ===== | ||

+ | |||

+ | {{f11finalq3fig.png}} | ||

+ | |||

+ | $\theta=\tan^{-1}\frac{30}{60}=26.57^{o}$ | ||

+ | |||

+ | Vertical forces, up is positive $T\sin\theta+Fh_{y}=m_{sign}g+m_{rod}g$ | ||

+ | |||

+ | Horizontal forces, right is positive $-T\cos\theta+Fh_{x}=0$ | ||

+ | |||

+ | Torques $0.6T\sin\theta=0.8m_{sign}g+0.4m_{rod}g$ | ||

+ | |||

+ | $T=\frac{0.8\times3\times9.8+0.4\times1\times9.8}{0.6\times\sin26.57^{o}}=102.24\mathrm{N}$ | ||

+ | |||

+ | $Fh_{y}=4.981-102.24\sin26.57^{o}=-6.53\mathrm{N}$ | ||

+ | |||

+ | So force points down. | ||

+ | |||

+ | $Fh_{x}=102.24\cos26.57^{o}=91.4\mathrm{N}$ | ||

+ | |||

+ | Force points to the right | ||

+ | ===== Will the ladder slip? ===== | ||

+ | |||

+ | {{ladderslip.png}} | ||

+ | |||

+ | The forces that act on the ladder of length $l$ are the weight of the ladder that acts down at it's center of mass, $m\vec{g}$, a normal force exerted by the wall, $\vec{f}_{NW}$, an normal force exerted by the ground $\vec{f}_{NG}$, and a force due to the friction of the ground, $\vec{F}_{Fr}$. The friction between the ladder and the wall is typically neglected. | ||

+ | |||

+ | For equilibrium the sum of the forces in both the horizontal and vertical directions must be zero and also the sum of the torques around an appropriate axis must be equal to zero. | ||

+ | |||

+ | Balance of the forces in the horizontal direction implies that | ||

+ | |||

+ | $F_{NW}=F_{Fr}$ | ||

+ | |||

+ | and in the vertical direction | ||

+ | |||

+ | $F_{NG}=m_{l}g$ | ||

+ | |||

+ | If we calculate the torques around the point at which the ladder touches the ground then they will be | ||

+ | |||

+ | $F_{Nw}l\sin\theta-m_{l}g\frac{l}{2} \sin(90^{o}-\theta)=0$ | ||

+ | |||

+ | $F_{Nw}l\sin\theta-m_{l}g\frac{l}{2}\cos\theta=0$ | ||

+ | |||

+ | This can be written in terms of the frictional force by using the equation we got from the horizontal forces. | ||

+ | |||

+ | $F_{fr}l\sin\theta-m_{l}g\frac{l}{2}\cos\theta=0$ | ||

+ | |||

+ | The maximum force that can be provided by friction is $\mu F_{NG}$ and so the maximum safe angle is when this frictional force is provided | ||

+ | |||

+ | $\mu F_{NG}l\sin\theta_{max}-m_{l}g\frac{l}{2}\cos\theta_{max}=0$ | ||

+ | |||

+ | $\mu mgl\sin\theta_{max}-m_{l}g\frac{l}{2}\cos\theta_{max}=0$ | ||

+ | |||

+ | $\tan\theta_{max}=\frac{1}{2\mu}$ | ||

+ | |||

+ | Of course if a person is standing on the ladder, their weight would need to be taken in to account both in the balance of the vertical forces and the torques. |