Respuesta :
ANSWER:
a) 8.68 m/s^2
b) 71.74 rad/s^2
c) 9.765 m
STEP-BY-STEP EXPLANATION:
Given:
m1 = 2.85 kg
m2 = 0.742 kg
r = 0.121 m
a)
From the Newton's second law of motion, the downward force acting on the bucket is:
[tex]\begin{gathered} \sum ^{}_{}F=m_1\cdot g-T=m_1\cdot a \\ T=m_1\cdot g-m_1\cdot a\text{ (1)} \end{gathered}[/tex]Torque acting on the pulley is:
[tex]\begin{gathered} \sum ^{}_{}\tau=I\alpha=rT \\ I\alpha=rT \end{gathered}[/tex]Hence, the angular acceleration is:
[tex]\begin{gathered} \alpha=\frac{a}{r},\text{ and inertia is, I }=\frac{1}{2}m_2\cdot r^2 \\ \text{ therefore:} \\ \frac{1}{2}m_2\cdot r^2\cdot\frac{a}{r}=rT \\ T=\frac{1}{2}m_2\cdot a\text{ (2)} \end{gathered}[/tex]We equate equations (1) and (2), and solve for the acceleration like this:
[tex]\begin{gathered} m_1\cdot g-m_1\cdot a=\frac{1}{2}m_2\cdot a \\ \frac{1}{2}m_2\cdot a+m_1\cdot a=m_1\cdot g \\ a\cdot(\frac{1}{2}m_2+m_1)=m_1\cdot g \\ a=\frac{m_1\cdot g}{\frac{1}{2}m_2+m_1} \\ \text{ replacing:} \\ a=\frac{2.85\cdot9.81}{\frac{1}{2}\cdot0.742+2.85} \\ a=8.68m/s^2 \end{gathered}[/tex]b)
The angular acceleration of the pulley is:
[tex]\begin{gathered} \alpha=\frac{a}{r} \\ \text{ replacing} \\ \alpha=\frac{8.68}{0.121} \\ \alpha=71.74rad/s^2 \end{gathered}[/tex]c)
We use the following equation to be able to calculate the distance:
[tex]\begin{gathered} x=\frac{1}{2}a\cdot t^2 \\ \text{ replacing} \\ x=\frac{1}{2}\cdot8.68\cdot(1.5)^2 \\ x=9.765\text{ m} \end{gathered}[/tex]