Question

A torque of magnitude 500Nm acts on a body of mass 16 and produces an angular acceleration of 20rad .the radius of gyration of the body

Answer 1

Answer:

Given:

Torque = 500 Nm

Mass = 16 kg.

Angular acceleration = 20 rad/s²

To find:

Radius of gyration of the body

Concept:

Radius of gyration is the radius of a circular ring having same Moment of Inertia as that of another object along a specified axis.

It is denoted by k

Calculation:

$\huge{ \green{ \tau = I \alpha }}$

$\huge{ \green{ = > 500 = I \times 20 }}$

$\huge{ \green{ = > I = 25 \: kg \: {m}^{2} }}$

Now , we know that :

$\huge{ \red{I = m {k}^{2} }}$

$\huge{ \red{ = > 25 = 16 {k}^{2} }}$

$\huge{ \red{ = > {k}^{2} = \frac{25}{16} }}$

$\huge{ \red{ = > k= \frac{5}{4} }}$

$\huge{ \red{ = > k= 1.25 \: m }}$

So final answer is :

$\boxed{ \huge{ \blue{ \sf{ k= 1.25 \: m }}}}$

Answer 2

Answer:

• The radius of gyration (K) is 1.25 meters.

Given:

1. Torque (τ) = 500 N m.
2. Mass of the body (M) = 16 Kg
3. Angular Acceleration (α) = 20 rad / sec.

Explanation:

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From the formula we Know,

⇒ τ = I α

Where,

• τ Denotes Torque.
• I Denotes Moment of Inertia.
• α Denotes Angular Acceleration.

Now,

⇒ τ = I α

Substituting the values,

⇒ 500 N/m = I × 20 rad/sec

⇒ 500  = I × 20

⇒ I = 500 / 20

⇒ I = 25

⇒ I = 25 Kg-m².

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From the relation We know,

⇒ I = M K²

Where,

• I Denotes Moment of Inertia.
• M Denotes Mass
• K Denotes Radius of Gyration.

Now,

⇒ I = M K²

Substituting the values,

⇒ 25 Kg-m² = 16 Kg × K²

⇒ 25 = 16 × K²

⇒ K² = 25 / 16

⇒ K = √ (25 / 16)

⇒ K = 5 / 4

⇒ K = 1.25

⇒ K = 1.25 m.

∴ The radius of gyration (K) is 1.25 meters.

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