Question

Write the dimensional formula of the ratio of linear momentum to angular momentum. (a) \(\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{0}\) (c) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{0}\) (d) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{1}\)

Short answer

The short answer is: The dimensional formula of the ratio of linear momentum to angular momentum is (a) \(\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0}\).

Step-by-step solution

Write the formula for linear momentum

Linear momentum is calculated by the product of the mass and velocity of the object: P = m*v, where P is the linear momentum, m is the mass, and v is the velocity.

Find the dimensional formula for linear momentum

To find the dimensional formula for linear momentum, we need to determine the dimensions of mass and velocity. Mass has the dimension M and velocity has dimensions of length L and time T (L/T). Therefore, the dimensional formula for linear momentum is: \[ [P] = [m] \times [v] = M \times \frac{L}{T} = M L T^{-1} \]

Write the formula for angular momentum

Angular momentum is calculated by the product of the moment of inertia and angular velocity of the object: L = I*ω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Find the dimensional formula for angular momentum

To find the dimensional formula for angular momentum, we need to determine the dimensions of the moment of inertia and angular velocity. The moment of inertia has dimensions of mass and length (ML²) and angular velocity has dimensions of time (T^(-1)). Therefore, the dimensional formula for angular momentum is: \[ [L] = [I] \times [\omega] = (M L^2) \times (T^{-1}) = M L^2 T^{-1} \]

Determine the dimensional formula for the ratio of linear momentum to angular momentum

To find the dimensional formula for the ratio of linear momentum to angular momentum, we need to divide the dimension of linear momentum by the dimension of angular momentum. \[ \frac{[P]}{[L]} = \frac{M L T^{-1}}{M L^2 T^{-1}} = M^0 L^{-1} T^0 \]

Match the result to one of the answer options

Comparing the result (\(M^0 L^{-1} T^0\)) to the options given, we can see that it matches with option (a). Therefore, the correct answer is (a) \(\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0}\).

Key concepts

Linear Momentum

Understanding linear momentum is a key aspect of physics. It's an important concept because it describes the motion of an object. Linear momentum is defined as the product of an object's mass and its velocity. Simply put, linear momentum, often represented as \(P\), measures how difficult it is to stop a moving object. The formula is given by:

• \(P = m \times v\)

where \(m\) is the mass and \(v\) is the velocity.

The dimensions of linear momentum are critical for understanding various physics problems. To find these dimensions, consider the mass \((M)\) and velocity, which itself is the ratio of length \((L)\) to time \((T)\). Therefore, the dimensions are:

• \([P] = M L T^{-1}\)

These dimensions show the characteristics of linear momentum that contribute to an object’s state of motion.

Angular Momentum

Angular momentum, much like linear momentum, is a measure of how much motion an object has, but in terms of rotation. It is crucial when studying systems that have rotational motion, such as planets orbiting a star or a spinning top. Angular momentum is the product of the moment of inertia and angular velocity:

• \(L = I \times \omega\)

where \(L\) is angular momentum, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity.

The dimension of angular momentum involves understanding both the moment of inertia and angular velocity. The moment of inertia, resembling a "rotational mass," has the dimension \(ML^2\), while angular velocity is the rate of change of angular position over time, with dimensions \(T^{-1}\). Thus, the dimension of angular momentum is:

• \([L] = M L^2 T^{-1}\)

This makes it crucial for analyzing systems in rotational equilibrium or dynamic motion.

Moment of Inertia

Moment of inertia is a fundamental property in rotational dynamics, often compared to mass in linear motion. It is a measure of an object's resistance to changes in its rotational motion. The moment of inertia depends on the distribution of mass around an axis of rotation.

In general terms, for a rigid body, the moment of inertia \(I\) is expressed as:

• \(I = \sum m_i r_i^2\)

Here, \(m_i\) represents an infinitesimal mass element and \(r_i\) its perpendicular distance from the axis of rotation.

In terms of dimensions, the moment of inertia incorporates mass \((M)\) and the square of length \((L^2)\), giving:

• \([I] = M L^2\)

This dimension signifies how mass and shape influence an object's ability to rotate and remain stable under rotational forces.