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Chapter 8: Problem 47

Verify that the units of the rotational form of Newton's second law [Eq. (8-9)] are consistent. In other words, show that the product of a rotational inertia expressed in \(\mathrm{kg} \cdot \mathrm{m}^{2}\) and an angular acceleration expressed in \(\mathrm{rad} / \mathrm{s}^{2}\) is a torque expressed in \(\mathrm{N} \cdot \mathrm{m}\).

Short Answer

Expert verified
Answer: Yes, the units of the rotational form of Newton's second law are consistent. The product of rotational inertia (\(\mathrm{kg} \cdot \mathrm{m}^{2}\)) and angular acceleration (\(\mathrm{rad}/\mathrm{s}^{2}\)) results in the same units as torque (\(\mathrm{N}\cdot\mathrm{m}\)), which confirms the consistency.

Step by step solution

01

Identify the units of rotational inertia, angular acceleration, and torque

Rotational Inertia (\(I\)): \(\mathrm{kg} \cdot \mathrm{m}^{2}\) Angular Acceleration (\(\alpha\)): \(\mathrm{rad} / \mathrm{s}^{2}\) Torque (\(\tau\)): \(\mathrm{N} \cdot \mathrm{m}\)
02

Express Newton's second law in rotational form

Newton's second law for rotation states that the torque is equal to the rotational inertia multiplied by the angular acceleration: \(\tau = I * \alpha\)
03

Check the units by multiplying the units of rotational inertia and angular acceleration

Now we perform the multiplication \(\mathrm{kg} \cdot \mathrm{m}^{2} * \dfrac{\mathrm{rad}}{\mathrm{s}^{2}}\). It's also important to know that \(\mathrm{rad}\) is a dimensionless unit, so it can be removed in the multiplication. \(\mathrm{kg} \cdot \mathrm{m}^{2} * \dfrac{\mathrm{1}}{\mathrm{s}^{2}} = \dfrac{\mathrm{kg} \cdot \mathrm{m}^{2}}{\mathrm{s}^{2}}\)
04

Convert the units of torque to the standard unit

We have to convert the unit of torque from \(\mathrm{N} \cdot \mathrm{m}\) to the standard units of mass, length, and time to compare it with the result obtained in Step 3. We know that \(\mathrm{N}\) (newton) is equal to \(\mathrm{kg} \cdot \mathrm{m}/\mathrm{s}^{2}\), so we can substitute it back: \(\mathrm{N} \cdot \mathrm{m} = \mathrm{(kg} \cdot \mathrm{m}/\mathrm{s}^{2}) \cdot \mathrm{m} = \dfrac{\mathrm{kg} \cdot \mathrm{m}^{2}}{\mathrm{s}^{2}}\)
05

Compare the results

Now we compare our final result from Step 3 and Step 4: \(\dfrac{\mathrm{kg} \cdot \mathrm{m}^{2}}{\mathrm{s}^{2}} = \dfrac{\mathrm{kg} \cdot \mathrm{m}^{2}}{\mathrm{s}^{2}}\) Since both units are the same, we can conclude that the units of the rotational form of Newton's second law are consistent.

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