From e6de45757757d447df542ef079a5bde3205c45c7 Mon Sep 17 00:00:00 2001 From: Krishna Date: Mon, 23 Sep 2024 22:13:09 -0500 Subject: [PATCH] circular motion so far --- content/physics/Forces.md | 33 +++++++++++++++++++++++++++- content/physics/circular-motion.png | Bin 0 -> 77309 bytes content/physics/perp-vectors.png | Bin 0 -> 35479 bytes 3 files changed, 32 insertions(+), 1 deletion(-) create mode 100644 content/physics/circular-motion.png create mode 100644 content/physics/perp-vectors.png diff --git a/content/physics/Forces.md b/content/physics/Forces.md index 49174f4..fb9ba45 100644 --- a/content/physics/Forces.md +++ b/content/physics/Forces.md @@ -203,7 +203,38 @@ $$ # Circular motion -**TODO!** +An object that moves in a circle at a constant speed $||\vec{v}||$ experiences **uniform circular motion**. The most important part of that sentence is the constant speed part; the measurement $||\vec{v}||$ is reference to the center of the circle. + +![](circular-motion.png) + +As seen above, a velocity for every point on the path of the circle's motion exits, and the change in the two velocity vectors from any two points separated by $\Delta t$ with respect to time computes the acceleration. As these two vectors become closer in size and more parallel, the difference becomes the vertical difference between the two showed in Figure 5-2 B. The acceleration vector is always directed to the center of the circle. +$$ +a_R=\lim_{\Delta t\to0}{\frac{\Delta \vec{v}}{\Delta t}}=\frac{v^2}{r} +$$ +$a_R$ is the radial acceleration, otherwise known as the centripetal acceleration. It's easier not to consider centripetal acceleration as a result of the imaginary centripetal force if instead thought of as acceleration experienced by the object radially, or on the radius of motion. +$$ +T=\frac{1}{f} +$$ + +$f$ is the revolutions per second of the object, usually expressed in its SI units: hertz. In radians: + +$$ +v=\frac{\text{distance}}{\text{time}}=\frac{2\pi r}{T} +$$ + +It's important to note that the acceleration and velocity vectors at an instant are always perpendicular. Additionally, in uniform circular motion, the magnitude of the acceleration vector is always constant. + +![](perp-vectors.png) + +> [!FAQ] Question +> A $150$ gram ball at the end of a string is revolving uniformly in a horizontal circle of radius $0.6$ m. The ball makes 2 revolutions in a second. What is its centripetal acceleration? + +> [!NOTE]- Answer +> First recognize the centripetal acceleration formula is $a_R=v^2/r$. $r$ is give. Additionally, we must use the velocity formula for fixed periods: \ +> $v=\frac{2\pi r}{T}$. \ +> Since the ball makes 2 revolutions in a second, the period $T$ must be $\tfrac{1}{2}$. $v=\frac{2\pi(0.6\mathrm{m})}{0.5}=7.54$, the unit of that output being meters/second. Substituting that value for the speed $v$ in the acceleration formula, acceleration is $94.7 \mathrm{m}/\mathrm{s}^2$ + + diff --git a/content/physics/circular-motion.png b/content/physics/circular-motion.png new file mode 100644 index 0000000000000000000000000000000000000000..b8a8fcefa0644d42655a3e7c7e9ac0e194d77ebd GIT binary patch literal 77309 zcmeAS@N?(olHy`uVBq!ia0y~yV0_QO!1|ekiGhKkXoFfN0|NtRfk$L90|Va?5N4dJ z%_q&kpuphi;uum9_hxRngUIpk2cAnYxm$OYd|(iCad2=nbhm73;8^70el$bGG)ZU6 z4b!j-9j4P(Yz%jC>11_1IKL@kLt{dOw8o;Aj*^!9@}JM9+|<5PSl&@+U;lRR%-PlE zzst&=ot-Vo;6Wd-PbI_Upm=Ln7gw)ac4nrc+Jp%cCe)+`bTDZ;F29_9W`^UGPmdoQ zY<_=lud%r~KR>_wp|7v6*Z=syIQ@0~g^Vo^54RWZj8RutkByDBn#;$Kk(XEZ=f}sh zv(5WuEI;J#k|1bal?{DAB|A)_=J9qLVXWah!|Nnk`eEi|o zl`AF&1|RO%{J(JT-n|PK0^-3SB;?D3gU#EvnK{jqG)^;^>C-e}`uF$u|9`uk@AC3_ z+sPC&6O#}BG=(A$q-|ckdiDQ*zw?utW_YZYnYn23;^5%m%QtR#_$W+dh}`&M5(i+9F& zdwW}2?tJ#_+4J-Br%#*q?$*}V@9yqCaz`m4J^lFiSdPs6{PpYC|Nrsv@#p8~=i5{U 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