Question

Dimensions of impulse are. (a) \(\mathrm{M}^{-1} \mathrm{~L}^{-1} \mathrm{~T}^{1}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{1}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\)

Short answer

The dimensions of impulse are \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}\). Therefore, the correct answer is (b).

Step-by-step solution

Formula for Impulse

The formula for impulse (I) is given by \(I = F \cdot \Delta t\) where \(F\) is the force and \(\Delta t\) is the time interval.

Dimensions of Force and Time

The dimensions of Force (F) are given by: \(\mathrm{[F]} = \mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) since force is a product of mass and acceleration, and acceleration has dimensions of \(\mathrm{L}^{1} \mathrm{~T}^{-2}\). The dimensions of Time (\(\Delta t\)) are: \(\mathrm{[\Delta t]} = \mathrm{T}^{1}\)

Find the Dimensions of Impulse

Since impulse is a product of force and time, we multiply the dimensions of force by the dimensions of time: \(\mathrm{[I]} = \mathrm{[F]} \times \mathrm{[\Delta t]} \) \(\mathrm{[I]} = (\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}) \times (\mathrm{T}^{1})\) By multiplying the exponents with similar bases, we find the dimensions of impulse: \(\mathrm{[I]} = \mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}\) Now, let's compare the found dimensions with the options given in the exercise: (a) \(\mathrm{M}^{-1} \mathrm{~L}^{-1} \mathrm{~T}^{1}\) (b) \(\boxed{\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{1}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) The correct answer is (b) because it matches the dimensions we found for impulse: \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}\).

Key concepts

Impulse in Physics

Impulse is a fascinating concept in physics. It describes the change in momentum of an object. Think of it as a burst of force applied over a certain time. When an object experiences force for a time, its velocity changes, which results in a change in momentum. That's where impulse comes into play. The formula for impulse is given by:

• \( I = F \cdot \Delta t \)

where:

• \( I \) is the impulse
• \( F \) is the force applied
• \( \Delta t \) is the time duration

Impulse can be visualized in everyday life experiences. For example, when you hit a ball with a bat, the bat applies a certain amount of force for a brief period. This changes the ball's momentum, resulting in impulse.

Dimensional Analysis

Dimensional analysis is a simple yet powerful method in physics. It helps us understand the fundamental nature of equations. By examining the dimensions of physical quantities, we can verify the correctness of physical equations and conversions. Each type of physical quantity, like velocity or force, has its own dimension. When conducting dimensional analysis for impulse, we start with force and time, since these are its components. The dimensions of force are \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\), and the dimensions of time are \(\mathrm{T}^{1}\). To find the dimensions of impulse, we multiply these dimensions:

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

This tells us that impulse is expressed in terms of mass, length, and time.

Force and Time Relationship

Understanding the relationship between force and time is key to grasping the concept of impulse. When a force is applied to an object for a certain time, it leads to a change in the object's motion. The product of force and time is what defines impulse. This relationship is crucial because:

• It helps describe how long a force is acting on an object, affecting its velocity.
• A strong force over a short period can have the same effect as a weaker force over a longer duration.

For example, consider a car crash. A large force is applied over a short time to stop the car. Similarly, a gentle push for a longer time can also stop a moving cart. The interplay between force and time explains real-world situations effectively and demonstrates how impulse adjusts momentum.