Let's solve the problem **step-step**. --- ## **Given** - \(\theta = 30^\circ\) - \(\dot{\theta} = 10^\circ/\text{s} = .1745\, \text{rad/s}\) *(convert degrees to radians: \(\frac{10 \pi}{180}\))* - \(l = .5\, \text{m}\) - \(\dot{l} = .2\, \text{m/s}\) - \(\ddot{l} = -.3\, \text{m/s}^2\) - \(\ddot{\theta} = \) (since \(\dot{\theta}\) is constant) --- ## **1. Position Vector of Point P** Relative to the origin \(O\): \[ \vec{r}_P = l \cos \theta\, \mathbf{i} + l \sin \theta\, \mathbf{j} \] --- ## **2. Velocity of P** Differentiate \(\vec{r}_P\) with respect to time: \[ \vec{v}_P = \frac{d\vec{r}_P}{dt} \] \[ \frac{d}{dt}(l \cos\theta) = \dot{l}\cos\theta - l\sin\theta\,\dot{\theta} \] \[ \frac{d}{dt}(l \sin\theta) = \dot{l}\sin\theta + l\cos\theta\,\dot{\theta} \] So, \[ \boxed{ \vec{v}_P = [\dot{l}\cos\theta - l\sin\theta\,\dot{\theta}]\, \mathbf{i} + [\dot{l}\sin\theta + l\cos\theta\,\dot{\theta}]\, \mathbf{j} } \] --- ### **Plug in the Values** - \(\cos 30^\circ = .866\) - \(\sin 30^\circ = .5\) - \(l = .5\) - \(\dot{l} = .2\) - \(\dot{\theta} = .1745\) #### **i-component:** \[ \dot{l}\cos\theta - l\sin\theta\,\dot{\theta} = (.2)(.866) - (.5)(.5)(.1745) = .1732 - .0436 = .1296 \] #### **j-component:** \[ \dot{l}\sin\theta + l\cos\theta\,\dot{\theta} = (.2)(.5) + (.5)(.866)(.1745) = .1 + .0755 = .1755 \] \[ \boxed{ \vec{v}_P = .1296\, \mathbf{i} + .1755\, \mathbf{j}\ \text{m/s} } \] #### **Magnitude of \(\vec{v}_P\):** \[ |\vec{v}_P| = \sqrt{(.1296)^2 + (.1755)^2} = \sqrt{.0168 + .0308} = \sqrt{.0476} = .218\ \text{m/s} \] --- ## **3. Acceleration of P** Differentiate \(\vec{v}_P\) with respect to time: \[ \vec{a}_P = \frac{d\vec{v}_P}{dt} \] Calculate each term (using product and chain rules): ### **i-component:** \[ \frac{d}{dt}[\dot{l}\cos\theta - l\sin\theta\,\dot{\theta}] \] Expand: - \(\frac{d}{dt}[\dot{l}\cos\theta] = \ddot{l}\cos\theta - \dot{l}\sin\theta\,\dot{\theta}\) - \(\frac{d}{dt}[l\sin\theta\,\dot{\theta}] = \dot{l}\sin\theta\,\dot{\theta} + l\cos\theta\,\dot{\theta}^2\) (*since \(\ddot{\theta} = \)*) So, \[ a_{P,x} = \ddot{l}\cos\theta - \dot{l}\sin\theta\,\dot{\theta} - \left[ \dot{l}\sin\theta\,\dot{\theta} + l\cos\theta\,\dot{\theta}^2 \right] \] \[ = \ddot{l}\cos\theta - 2\dot{l}\sin\theta\,\dot{\theta} - l\cos\theta\,\dot{\theta}^2 \] ### **j-component:** \[ \frac{d}{dt}[\dot{l}\sin\theta + l\cos\theta\,\dot{\theta}] \] Expand: - \(\frac{d}{dt}[\dot{l}\sin\theta] = \ddot{l}\sin\theta + \dot{l}\cos\theta\,\dot{\theta}\) - \(\frac{d}{dt}[l\cos\theta\,\dot{\theta}] = \dot{l}\cos\theta\,\dot{\theta} - l\sin\theta\,\dot{\theta}^2\) So, \[ a_{P,y} = \ddot{l}\sin\theta + 2\dot{l}\cos\theta\,\dot{\theta} - l\sin\theta\,\dot{\theta}^2 \] --- ### **Plug in the Values** \[ \ddot{l} = -.3,\, \cos\theta = .866,\, \sin\theta = .5,\, \dot{l} = .2,\, \dot{\theta} = .1745,\, l = .5 \] #### **i-component:** \[ a_{P,x} = (-.3)(.866) - 2(.2)(.5)(.1745) - .5(.866)(.1745^2) \] \[ = -.2598 - .0349 - .0132 \] \[ = -.3079 \] #### **j-component:** \[ a_{P,y} = (-.3)(.5) + 2(.2)(.866)(.1745) - .5(.5)(.1745^2) \] \[ = -.15 + .0604 - .0076 \] \[ = -.0972 \] \[ \boxed{ \vec{a}_P = -.3079\, \mathbf{i} - .0972\, \mathbf{j}\ \text{m/s}^2 } \] #### **Magnitude of \(\vec{a}_P\):** \[ |\vec{a}_P| = \sqrt{(-.3079)^2 + (-.0972)^2} = \sqrt{.0948 + .00945} = \sqrt{.1042} = .323\ \text{m/s}^2 \] --- ## **Final Answers** ### **(a) Velocity (in unit vectors):** \[ \boxed{ \vec{v}_P = .1296\, \mathbf{i} + .1755\, \mathbf{j}\ \text{m/s} } \] ### **(b) Velocity Magnitude:** \[ \boxed{ |\vec{v}_P| = .218\ \text{m/s} } \] ### **(c) Acceleration (in unit vectors):** \[ \boxed{ \vec{a}_P = -.3079\, \mathbf{i} - .0972\, \mathbf{j}\ \text{m/s}^2 } \] ### **(d) Acceleration Magnitude:** \[ \boxed{ |\vec{a}_P| = .323\ \text{m/s}^2 } \] --- **Let me know if you need clarification on any step!**