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Circular Motion


Mahi Tuatahi

Aaron is painting the outside of his house. He is standing on a $3.5m$ long plank with a support at each end. The plank weighs $4.8kg$. He is standing $0.8m$ from the left side and he weighs $63kg$.

  1. Draw a diagram to illustrate the situation
  2. Calculate the support force provided by Support A (left) and Support B (right).

Circular Motion

The motion of an object moving in a circular path.

e.g. Satellites in orbit, car driving around a corner, discus thrower, cricket bowler.


Circles

\begin{aligned} & Radius = r \newline & Diameter = d \newline & Circumference = C \end{aligned}


Pātai Tahi (Q1): Circumference


Pātai Rua (Q2): What is Frequency?


Pātai Toru (Q3): What is Period?


Velocity on a Circle


Pātai Whā (Q4): Finding Velocity

  1. If the radius is 2m, find the: circumference,
  2. and speed

Whakatika

  1. If the radius is 2m, find the: circumference,
    • $C = 2 \pi r = 2 \pi 2 = 12.57m$
  2. period,
    • $T = 12s$
  3. frequency
    • $f = \frac{1}{T} = \frac{1}{12} = 0.083^{-s}$
  4. and speed
    • $v = \frac{d}{t} = \frac{12.57}{12} = 1.0475ms^{-1}$

Pātai Rimu (Q5)


Whakatika


Centripetal Acceleration

An object undergoing circular motion is always accelerating towards the center of the circle. Therefore, because the direction is changing, the velocity is changing. Therefore the object is always accelerating, even if its speed is constant.

\begin{aligned} & a_{c} = \frac{v^{2}}{r} \newline \end{aligned}


Mahi Tuatahi

  1. An ice skater performs a 720 degree jump. The outside of their shoulders rotate with a frequency of $1.5s^{-1}$. Calculate the period of their rotations.
  2. If their shoulders are $30cm$ from their center of rotation, calculate their linear velocity.
  3. Draw a diagram illustrating the things you know about their circle of rotation.

Whakatika

  1. __An ice skater performs a 720 degree jump. The outside of their shoulders rotate with a frequency of $1.5s^{-1}$. Calculate the period of their rotations.__
    $T = \frac{1}{f} = \frac{1}{1.5} = 0.667s$
  2. __If their shoulders are $30cm$ from their center of rotation, calculate their linear velocity.__
    $v=\frac{2 \pi r}{T} = \frac{2 \pi 0.3}{0.667} = 2.826ms^{-1}$
  3. Draw a diagram illustrating the things you know about their circle of rotation.


What Causes the Acceleration?


Centripetal Force

\begin{aligned} & F_{c} = \frac{mv^{2}}{r} \end{aligned}



Pātai Ono (Q6): Bucket of Water


Whakatika

\begin{aligned} & v = 4ms^{-1}, r = 1m, m = 8kg && \text{(K)} \newline & F_{c} = ? && \text{(U)} \newline & F_{c} = \frac{mv^{2}}{r} && \text{(F)} \newline & F_{c} = \frac{8 \times 4^{2}}{1} = 128N \text{ inwards} && \text{(S+S)} \end{aligned}


NB: From 2:30 to 5:25.


Pātai Whitu (Q7): Hammer Throw


Whakatika

\begin{aligned} & F_{c} = 350N, v = 10ms^{-1}, r = 2m && \text{(K)} \newline & m = ? && \text{(U)} \newline & F_{c} = \frac{mv^{2}}{r} && \text{(F)} \newline & 350 = \frac{m \times 10^{2}}{2} && \text{(S+S)} \newline & \frac{350 \times 2}{10^{2}} = m = 7kg \end{aligned}


Pātai Waru (Q8): Exam Question 26-28