A rotating shaft with a flywheel exhibits torsional vibration at its natural frequency. Given torsional stiffness (k_t) = 1,510 lbf-ft/rad; mass moment of inertia (I) = 24.6000 slug-ft^2, determine the natural circular frequency (omega_n) in rad/s.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• Everything except omega_n is given, so isolate omega_n 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for omega_n:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{I}}$$

3. Step 3 — List the givens: torsional stiffness (k_t) = 1,510 lbf-ft/rad, mass moment of inertia (I) = 24.6000 slug-ft^2.

4. Step 4 — Substitute the given values:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{24.6000}}$$

5. Step 5 — Evaluate:

   $$\omega_{n} = 7.8347\ \text{rad/s}$$

6. Step 6 — Check: returning omega_n = 7.8347 rad/s to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 15.6693 — kept a factor of two that cancels in the correct rearrangement.
• 3.9173 — dropped that same factor in the other direction.
• 8.6181 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

An engine crankshaft experiences torsional vibration during operation. Given mass moment of inertia (I) = 24.4000 slug-ft^2; natural circular frequency (omega_n) = 36.7000 rad/s, determine the torsional stiffness (k_t) in lbf-ft/rad.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• Everything except k_t is given, so isolate k_t 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for k_t:

   $$k_{t} = \omega_n^2 I$$

3. Step 3 — List the givens: mass moment of inertia (I) = 24.4000 slug-ft^2, natural circular frequency (omega_n) = 36.7000 rad/s.

4. Step 4 — Substitute the given values:

   $$k_{t} = \omega_n^2 24.4000$$

5. Step 5 — Evaluate:

   $$k_{t} = 32864\ \text{lbf-ft/rad}$$

6. Step 6 — Check: returning k_t = 32,864 lbf-ft/rad to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 65,728 — kept a factor of two that cancels in the correct rearrangement.
• 16,432 — dropped that same factor in the other direction.
• 36,151 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

A drive shaft coupling is checked for torsional vibration resonance. Given torsional stiffness (k_t) = 1,710 lbf-ft/rad; natural circular frequency (omega_n) = 184.6 rad/s, determine the mass moment of inertia (I) in slug-ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for I:

   $$I = \dfrac{k_t}{\omega_n^2}$$

3. Step 3 — List the givens: torsional stiffness (k_t) = 1,710 lbf-ft/rad, natural circular frequency (omega_n) = 184.6 rad/s.

4. Step 4 — Substitute the given values:

   $$I = \dfrac{k_t}{\omega_n^2}$$

5. Step 5 — Evaluate:

   I = 0.0502\ \text{slug-ft^2}

6. Step 6 — Check: returning I = 0.0502 slug-ft^2 to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 0.1004 — kept a factor of two that cancels in the correct rearrangement.
• 0.0251 — dropped that same factor in the other direction.
• 0.0552 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

A rotating shaft with a flywheel exhibits torsional vibration at its natural frequency. Given torsional stiffness (k_t) = 4,430 lbf-ft/rad; mass moment of inertia (I) = 6.4000 slug-ft^2, determine the natural circular frequency (omega_n) in rad/s.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• Everything except omega_n is given, so isolate omega_n 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for omega_n:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{I}}$$

3. Step 3 — List the givens: torsional stiffness (k_t) = 4,430 lbf-ft/rad, mass moment of inertia (I) = 6.4000 slug-ft^2.

4. Step 4 — Substitute the given values:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{6.4000}}$$

5. Step 5 — Evaluate:

   $$\omega_{n} = 26.3095\ \text{rad/s}$$

6. Step 6 — Check: returning omega_n = 26.3095 rad/s to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 52.6189 — kept a factor of two that cancels in the correct rearrangement.
• 13.1547 — dropped that same factor in the other direction.
• 28.9404 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

An engine crankshaft experiences torsional vibration during operation. Given mass moment of inertia (I) = 5.1000 slug-ft^2; natural circular frequency (omega_n) = 179.7 rad/s, determine the torsional stiffness (k_t) in lbf-ft/rad.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• Everything except k_t is given, so isolate k_t 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for k_t:

   $$k_{t} = \omega_n^2 I$$

3. Step 3 — List the givens: mass moment of inertia (I) = 5.1000 slug-ft^2, natural circular frequency (omega_n) = 179.7 rad/s.

4. Step 4 — Substitute the given values:

   $$k_{t} = \omega_n^2 5.1000$$

5. Step 5 — Evaluate:

   $$k_{t} = 164690\ \text{lbf-ft/rad}$$

6. Step 6 — Check: returning k_t = 164,690 lbf-ft/rad to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 329,379 — kept a factor of two that cancels in the correct rearrangement.
• 82,345 — dropped that same factor in the other direction.
• 181,159 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

A drive shaft coupling is checked for torsional vibration resonance. Given torsional stiffness (k_t) = 2,420 lbf-ft/rad; natural circular frequency (omega_n) = 137.6 rad/s, determine the mass moment of inertia (I) in slug-ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for I:

   $$I = \dfrac{k_t}{\omega_n^2}$$

3. Step 3 — List the givens: torsional stiffness (k_t) = 2,420 lbf-ft/rad, natural circular frequency (omega_n) = 137.6 rad/s.

4. Step 4 — Substitute the given values:

   $$I = \dfrac{k_t}{\omega_n^2}$$

5. Step 5 — Evaluate:

   I = 0.1278\ \text{slug-ft^2}

6. Step 6 — Check: returning I = 0.1278 slug-ft^2 to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 0.2556 — kept a factor of two that cancels in the correct rearrangement.
• 0.0639 — dropped that same factor in the other direction.
• 0.1406 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

A rotating shaft with a flywheel exhibits torsional vibration at its natural frequency. Given torsional stiffness (k_t) = 3,030 lbf-ft/rad; mass moment of inertia (I) = 33.5000 slug-ft^2, determine the natural circular frequency (omega_n) in rad/s.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• Everything except omega_n is given, so isolate omega_n 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for omega_n:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{I}}$$

3. Step 3 — List the givens: torsional stiffness (k_t) = 3,030 lbf-ft/rad, mass moment of inertia (I) = 33.5000 slug-ft^2.

4. Step 4 — Substitute the given values:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{33.5000}}$$

5. Step 5 — Evaluate:

   $$\omega_{n} = 9.5104\ \text{rad/s}$$

6. Step 6 — Check: returning omega_n = 9.5104 rad/s to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 19.0208 — kept a factor of two that cancels in the correct rearrangement.
• 4.7552 — dropped that same factor in the other direction.
• 10.4614 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

An engine crankshaft experiences torsional vibration during operation. Given mass moment of inertia (I) = 39.4000 slug-ft^2; natural circular frequency (omega_n) = 58.6000 rad/s, determine the torsional stiffness (k_t) in lbf-ft/rad.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• Everything except k_t is given, so isolate k_t 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for k_t:

   $$k_{t} = \omega_n^2 I$$

3. Step 3 — List the givens: mass moment of inertia (I) = 39.4000 slug-ft^2, natural circular frequency (omega_n) = 58.6000 rad/s.

4. Step 4 — Substitute the given values:

   $$k_{t} = \omega_n^2 39.4000$$

5. Step 5 — Evaluate:

   $$k_{t} = 135298\ \text{lbf-ft/rad}$$

6. Step 6 — Check: returning k_t = 135,298 lbf-ft/rad to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 270,596 — kept a factor of two that cancels in the correct rearrangement.
• 67,649 — dropped that same factor in the other direction.
• 148,828 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

A drive shaft coupling is checked for torsional vibration resonance. Given torsional stiffness (k_t) = 2,110 lbf-ft/rad; natural circular frequency (omega_n) = 75.4000 rad/s, determine the mass moment of inertia (I) in slug-ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for I:

   $$I = \dfrac{k_t}{\omega_n^2}$$

3. Step 3 — List the givens: torsional stiffness (k_t) = 2,110 lbf-ft/rad, natural circular frequency (omega_n) = 75.4000 rad/s.

4. Step 4 — Substitute the given values:

   $$I = \dfrac{k_t}{\omega_n^2}$$

5. Step 5 — Evaluate:

   I = 0.3711\ \text{slug-ft^2}

6. Step 6 — Check: returning I = 0.3711 slug-ft^2 to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 0.7423 — kept a factor of two that cancels in the correct rearrangement.
• 0.1856 — dropped that same factor in the other direction.
• 0.4083 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration

A rotating shaft with a flywheel exhibits torsional vibration at its natural frequency. Given torsional stiffness (k_t) = 3,010 lbf-ft/rad; mass moment of inertia (I) = 31.2000 slug-ft^2, determine the natural circular frequency (omega_n) in rad/s.

Start with the thinking

• The governing relation printed in this handbook section is Torsional vibration natural frequency.
• Everything except omega_n is given, so isolate omega_n 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.
• Torsional vibration of a shaft-disk system depends on the torsional stiffness and the disk's mass moment of inertia.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

2. Step 2 — Rearrange symbolically for omega_n:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{I}}$$

3. Step 3 — List the givens: torsional stiffness (k_t) = 3,010 lbf-ft/rad, mass moment of inertia (I) = 31.2000 slug-ft^2.

4. Step 4 — Substitute the given values:

   $$\omega_{n} = \sqrt{\dfrac{k_t}{31.2000}}$$

5. Step 5 — Evaluate:

   $$\omega_{n} = 9.8221\ \text{rad/s}$$

6. Step 6 — Check: returning omega_n = 9.8221 rad/s to

   $$\omega_n = \sqrt{\dfrac{k_t}{I}}$$

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

Why the other options are there

• 19.6443 — kept a factor of two that cancels in the correct rearrangement.
• 4.9111 — dropped that same factor in the other direction.
• 10.8043 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Torsional Vibration