The quantities which corresponds in linear motion to the quantities I, `vecJ, vectau and vecomega` in rotatory motion are respectively.

Options

A. `M, vecp, vecF and vecv`
B. `m, vecF, vecv and vecp`
C. `M, vecp, vecv and vecF`
D. `eta, vec tau, vecp and vec tau`

Answer

To solve the question regarding the correspondence of quantities in linear motion to those in rotational motion, we will identify each rotational quantity and its linear counterpart step by step.

### Step-by-Step Solution:

1. **Identify the Rotational Quantity: Moment of Inertia (I)**
- In rotational motion, the moment of inertia (I) is a measure of an object's resistance to changes in its rotation.
- **Corresponding Linear Quantity:** Mass (m)
- **Explanation:** Just as mass measures an object's resistance to changes in its linear motion, moment of inertia measures resistance to changes in rotational motion.

2. **Identify the Rotational Quantity: Angular Momentum (J vector)**
- Angular momentum (J vector) is the rotational equivalent of linear momentum and is a measure of the amount of rotational motion an object has.
- **Corresponding Linear Quantity:** Linear Momentum (p vector)
- **Explanation:** Linear momentum is defined as the product of mass and velocity, while angular momentum is defined as the product of moment of inertia and angular velocity.

3. **Identify the Rotational Quantity: Torque (tau vector)**
- Torque (tau vector) is the rotational equivalent of force and is a measure of the rotational effect of a force applied at a distance from the axis of rotation.
- **Corresponding Linear Quantity:** Force (F vector)
- **Explanation:** Just as force causes linear acceleration, torque causes angular acceleration.

4. **Identify the Rotational Quantity: Angular Velocity (omega vector)**
- Angular velocity (omega vector) describes how quickly an object is rotating.
- **Corresponding Linear Quantity:** Linear Velocity (v vector)
- **Explanation:** Linear velocity measures how quickly an object is moving in a straight line, while angular velocity measures how quickly it is rotating.

### Summary of Correspondences:
- Moment of Inertia (I) → Mass (m)
- Angular Momentum (J vector) → Linear Momentum (p vector)
- Torque (tau vector) → Force (F vector)
- Angular Velocity (omega vector) → Linear Velocity (v vector)