36.2: Problems
- Page ID
- 92271
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Translational Problem
Consider the following translational problem: a body of mass \(m=3.0 \mathrm{~kg}\) is initally at rest; then a force of \(F=5.0 \mathrm{~N}\) is applied to it for time \(t=7.0\) seconds. What is the final velocity \(v\) of the body?
Solution. Given the force, we can find the acceleration; knowing the acceleration and time, we can find the velocity. The applicable equations are
\[F =m a \]
\[v =a t+v_{0} .\]
Solving Eq. \(\PageIndex{1}\) for \(a\) and substituting into Eq. \(\PageIndex{2}\), we have
\[v=\left(\frac{F}{m}\right) t+v_{0}\]
Substituting the given values of \(F, m\), and \(t\), and using \(v_{0}=0\), we have
\[v=\left(\frac{5.0 \mathrm{~N}}{3.0 \mathrm{~kg}}\right)(7.0 \mathrm{~s})\]
or
\[v=11.67 \mathrm{~m} / \mathrm{s}\]
Rotational Problem
Now consider the following similar rotational problem, which can be solved using the same method: a body of moment of inertia \(I=3.0 \mathrm{~kg} \mathrm{~m}^{2}\) is initially at rest (not rotating); then a torque of \(\tau=5.0 \mathrm{~N} \mathrm{~m}\) is applied to it for time \(t=7.0\) seconds. What is the final angular velocity \(\omega\) of the body?
Solution. Given the torque, we can find the angular acceleration; knowing the angular acceleration and time, we can find the angular velocity. The applicable equations are analogous to those used for the translational problem:
\[\tau =I \alpha \]
\[\omega =\alpha t+\omega_{0} .\]
Solving Eq. \(\PageIndex{6}\) for \(\alpha\) and substituting into Eq. \(\PageIndex{7}\), we have
\[\omega=\left(\frac{\tau}{I}\right) t+\omega_{0}\]
Substituting the given values of \(\tau, I\), and \(t\), and using \(\omega_{0}=0\), we have
\[\omega=\left(\frac{5.0 \mathrm{~N} \mathrm{~m}}{3.0 \mathrm{~kg} \mathrm{~m}}\right)(7.0 \mathrm{~s})\]
or
\[\omega=11.67 \mathrm{rad} / \mathrm{s}\]