A flywheel's mass radius of gyration is used to find its mass moment of inertia. Given mass (m) = 18.1000 slug; radius of gyration (k) = 1.4500 ft, determine the mass moment of inertia (I) in slug-ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for I:

   $$I = m k^2$$

3. Step 3

   $$List the givens: mass (m) = 18.1000 slug, radius of gyration (k) = 1.4500 ft$$

4. Step 4 — Substitute the given values:

   $$I = 18.1000 1.4500^2$$

5. Step 5 — Evaluate:

   I = 38.0553\ \text{slug-ft^2}

6. Step 6 — Check: returning I = 38.0553 slug-ft^2 to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 76.1105 — kept a factor of two that cancels in the correct rearrangement.
• 19.0276 — dropped that same factor in the other direction.
• 41.8608 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A rotating disk's mass radius of gyration is computed from test data. Given radius of gyration (k) = 1.3500 ft; mass moment of inertia (I) = 85.6000 slug-ft^2, determine the mass (m) in slug.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except m is given, so isolate m symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for m:

   $$m = \dfrac{I}{k^2}$$

3. Step 3 — List the givens: radius of gyration (k) = 1.3500 ft, mass moment of inertia (I) = 85.6000 slug-ft^2.

4. Step 4 — Substitute the given values:

   $$m = \dfrac{85.6000}{1.3500^2}$$

5. Step 5 — Evaluate:

   $$m = 46.9684\ \text{slug}$$

6. Step 6 — Check: returning m = 46.9684 slug to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 93.9369 — kept a factor of two that cancels in the correct rearrangement.
• 23.4842 — dropped that same factor in the other direction.
• 51.6653 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A vehicle wheel's mass radius of gyration is estimated for a dynamics problem. Given mass (m) = 13.7000 slug; mass moment of inertia (I) = 90.9000 slug-ft^2, determine the radius of gyration (k) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \sqrt{\dfrac{I}{m}}$$

3. Step 3

   $$List the givens: mass (m) = 13.7000 slug, mass moment of inertia (I) = 90.9000 slug-ft^2$$

4. Step 4 — Substitute the given values:

   $$k = \sqrt{\dfrac{90.9000}{13.7000}}$$

5. Step 5 — Evaluate:

   $$k = 2.5759\ \text{ft}$$

6. Step 6 — Check: returning k = 2.5759 ft to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 5.1517 — kept a factor of two that cancels in the correct rearrangement.
• 1.2879 — dropped that same factor in the other direction.
• 2.8334 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A flywheel's mass radius of gyration is used to find its mass moment of inertia. Given mass (m) = 15.0000 slug; radius of gyration (k) = 1.4500 ft, determine the mass moment of inertia (I) in slug-ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for I:

   $$I = m k^2$$

3. Step 3

   $$List the givens: mass (m) = 15.0000 slug, radius of gyration (k) = 1.4500 ft$$

4. Step 4 — Substitute the given values:

   $$I = 15.0000 1.4500^2$$

5. Step 5 — Evaluate:

   I = 31.5375\ \text{slug-ft^2}

6. Step 6 — Check: returning I = 31.5375 slug-ft^2 to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 63.0750 — kept a factor of two that cancels in the correct rearrangement.
• 15.7688 — dropped that same factor in the other direction.
• 34.6913 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A rotating disk's mass radius of gyration is computed from test data. Given radius of gyration (k) = 1.8000 ft; mass moment of inertia (I) = 182.0 slug-ft^2, determine the mass (m) in slug.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except m is given, so isolate m symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for m:

   $$m = \dfrac{I}{k^2}$$

3. Step 3 — List the givens: radius of gyration (k) = 1.8000 ft, mass moment of inertia (I) = 182.0 slug-ft^2.

4. Step 4 — Substitute the given values:

   $$m = \dfrac{182.0}{1.8000^2}$$

5. Step 5 — Evaluate:

   $$m = 56.1728\ \text{slug}$$

6. Step 6 — Check: returning m = 56.1728 slug to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 112.3 — kept a factor of two that cancels in the correct rearrangement.
• 28.0864 — dropped that same factor in the other direction.
• 61.7901 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A vehicle wheel's mass radius of gyration is estimated for a dynamics problem. Given mass (m) = 27.4000 slug; mass moment of inertia (I) = 126.9 slug-ft^2, determine the radius of gyration (k) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \sqrt{\dfrac{I}{m}}$$

3. Step 3

   $$List the givens: mass (m) = 27.4000 slug, mass moment of inertia (I) = 126.9 slug-ft^2$$

4. Step 4 — Substitute the given values:

   $$k = \sqrt{\dfrac{126.9}{27.4000}}$$

5. Step 5 — Evaluate:

   $$k = 2.1521\ \text{ft}$$

6. Step 6 — Check: returning k = 2.1521 ft to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 4.3041 — kept a factor of two that cancels in the correct rearrangement.
• 1.0760 — dropped that same factor in the other direction.
• 2.3673 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A flywheel's mass radius of gyration is used to find its mass moment of inertia. Given mass (m) = 20.1000 slug; radius of gyration (k) = 1.1500 ft, determine the mass moment of inertia (I) in slug-ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for I:

   $$I = m k^2$$

3. Step 3

   $$List the givens: mass (m) = 20.1000 slug, radius of gyration (k) = 1.1500 ft$$

4. Step 4 — Substitute the given values:

   $$I = 20.1000 1.1500^2$$

5. Step 5 — Evaluate:

   I = 26.5823\ \text{slug-ft^2}

6. Step 6 — Check: returning I = 26.5823 slug-ft^2 to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 53.1645 — kept a factor of two that cancels in the correct rearrangement.
• 13.2911 — dropped that same factor in the other direction.
• 29.2405 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A rotating disk's mass radius of gyration is computed from test data. Given radius of gyration (k) = 2.1500 ft; mass moment of inertia (I) = 53.8000 slug-ft^2, determine the mass (m) in slug.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except m is given, so isolate m symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for m:

   $$m = \dfrac{I}{k^2}$$

3. Step 3 — List the givens: radius of gyration (k) = 2.1500 ft, mass moment of inertia (I) = 53.8000 slug-ft^2.

4. Step 4 — Substitute the given values:

   $$m = \dfrac{53.8000}{2.1500^2}$$

5. Step 5 — Evaluate:

   $$m = 11.6387\ \text{slug}$$

6. Step 6 — Check: returning m = 11.6387 slug to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 23.2774 — kept a factor of two that cancels in the correct rearrangement.
• 5.8194 — dropped that same factor in the other direction.
• 12.8026 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A vehicle wheel's mass radius of gyration is estimated for a dynamics problem. Given mass (m) = 6.4000 slug; mass moment of inertia (I) = 113.4 slug-ft^2, determine the radius of gyration (k) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \sqrt{\dfrac{I}{m}}$$

3. Step 3

   $$List the givens: mass (m) = 6.4000 slug, mass moment of inertia (I) = 113.4 slug-ft^2$$

4. Step 4 — Substitute the given values:

   $$k = \sqrt{\dfrac{113.4}{6.4000}}$$

5. Step 5 — Evaluate:

   $$k = 4.2094\ \text{ft}$$

6. Step 6 — Check: returning k = 4.2094 ft to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 8.4187 — kept a factor of two that cancels in the correct rearrangement.
• 2.1047 — dropped that same factor in the other direction.
• 4.6303 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration

A flywheel's mass radius of gyration is used to find its mass moment of inertia. Given mass (m) = 17.0000 slug; radius of gyration (k) = 0.5000 ft, determine the mass moment of inertia (I) in slug-ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Mass radius of gyration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The mass radius of gyration relates a rigid body's mass moment of inertia to an equivalent concentrated mass location.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I = m k^2$$

2. Step 2 — Rearrange symbolically for I:

   $$I = m k^2$$

3. Step 3

   $$List the givens: mass (m) = 17.0000 slug, radius of gyration (k) = 0.5000 ft$$

4. Step 4 — Substitute the given values:

   $$I = 17.0000 0.5000^2$$

5. Step 5 — Evaluate:

   I = 4.2500\ \text{slug-ft^2}

6. Step 6 — Check: returning I = 4.2500 slug-ft^2 to

   $$I = m k^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 8.5000 — kept a factor of two that cancels in the correct rearrangement.
• 2.1250 — dropped that same factor in the other direction.
• 4.6750 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Mass Radius of Gyration