Question

Dimensional formula for torque is (a) \(\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-3}\) (b) \(\mathrm{M}^{2} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\)

Short answer

The correct dimensional formula for torque is \(\mathrm{[\tau]} = \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\).

Step-by-step solution

Recall the formula for torque

Torque (\(\tau\)) can be defined as the cross product of force (\(\vec{F}\)) and perpendicular distance (\(\vec{r}\)) from the point of rotation. Mathematically, this is expressed as: \[\tau = |\vec{F}| \cdot |\vec{r}| \cdot sin(\theta)\]

Find the dimensional formula for force

According to Newton's second law, force (\(F\)) is the product of mass (\(M\)) and acceleration (\(a\)), which can be expressed as: \[F = M \cdot a\] Acceleration has the units of \(\mathrm{L} \mathrm{~T}^{-2}\), where L represents the length dimension, and T represents the time dimension. So, the dimensional formula for force (F) is given by: \[\mathrm{[F]} = \mathrm{M}^{1}\mathrm{~L}^{1}\mathrm{~T}^{-2}\]

Find the dimensional formula for distance

Distance (r) is a measure of length, which has the dimensional formula of L. Therefore, the dimensional formula for distance (r) is: \[\mathrm{[r]} = \mathrm{L}^{1}\]

Derive the dimensional formula for torque

Now, we can find the dimensional formula for torque (\(\tau\)) using the dimensional formulas for force (\(F\)) and distance (\(r\)). Since torque is the product of force and distance, we have: \[\mathrm{[\tau]} = \mathrm{[F]} \cdot \mathrm{[r]}\] Using the dimensional formulas we found earlier for force and distance: \[\mathrm{[\tau]} = \mathrm{M}^{1}\mathrm{~L}^{1}\mathrm{~T}^{-2} \cdot \mathrm{L}^{1}\] By combining the dimensions, we get: \[\mathrm{[\tau]} = \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\]

Compare with the given options and find the correct answer

Now, we can compare the derived dimensional formula for torque (\(\mathrm{[\tau]} = \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\)) with the given options: (a) \(\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-3}\) (b) \(\mathrm{M}^{2} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) We can clearly see that the derived dimensional formula for torque matches option (d). Therefore, the correct dimensional formula for torque is: \(\boxed{\mathrm{[\tau]} = \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}}\)

Key concepts

Torque

Torque is an important concept in physics that describes the rotational effect of a force applied to an object. You can think of torque as a measure of how effective a force is at making something rotate. When you open a door, you push it at some distance from the hinges. This distance, combined with the force of your push and the angle at which you apply it, determines how easily the door will swing open.

The mathematical expression of torque (\(\tau\)) is given by the cross product of the radius vector (\(\vec{r}\)) from the point of rotation to the point of force application, and the force vector (\(\vec{F}\)). The formula can be written as:

• \[ \tau = |\vec{F}| \cdot |\vec{r}| \cdot \sin(\theta) \]

Here, \(\theta\) is the angle between the radius and force vectors. The components involved are:

• Force (\(F\)): The pushing or pulling action applied to the object.
• Distance (\(r\)): The perpendicular distance from the pivot point to the line of action of the force.
• Angle (\(\theta\)): The angle between the direction of the force applied and the line connecting the pivot.

Understanding these concepts helps us determine how forces cause rotations in mechanical systems.

Newton's Second Law

Newton's Second Law is fundamental in understanding how force impacts the motion of an object. It states that the force applied to an object is equal to the mass of that object multiplied by its acceleration. The relationship can be formulated as:

• \[ F = M \cdot a \]

Where:

• \(F\) is the net force acting on the object.
• \(M\) is the mass of the object.
• \(a\) is the acceleration produced.

The law implies that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This concept is crucial for finding the dimensional formula for force, which is foundational to understanding torque further. Since acceleration itself has the dimensions of \(\mathrm{L} \mathrm{~T}^{-2}\) (with L for length and T for time), it helps derive that force carries dimensions of \(\mathrm{M}^{1}\mathrm{~L}^{1}\mathrm{~T}^{-2}\).

Dimensional Formula

Dimensional analysis is a powerful method used to find the dimensions of a physical quantity in terms of the basic physical dimensions such as mass (M), length (L), and time (T). A dimensional formula shows us how a physical quantity relates to the fundamental dimensions.

When we talk about torque, its dimensional formula can be derived by considering the product of dimensions for force and distance. Using dimensional analysis, we combined them as:

• Force (\([F] = \mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\))
• Distance (\([r] = \mathrm{L}^{1}\))

By multiplying these, we arrive at the dimensional formula for torque. Thus:

• \[\mathrm{[\tau]} = \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\]

This tells us that torque involves mass, length squared, and inverse time squared dimensions. Recognizing the proper dimensional formula is crucial for ensuring that calculations correspond correctly to real-world physical phenomena.