Question

The linear momentum \(\mathrm{P}\) of a particle varies with the time as follows. \(P=a+b t^{2}\) Where \(a\) and \(b\) are constants. The net force acting on the particle is (A) Proportional to t (B) Proportional to t \(^{2}\) (C) Zero (D) constant

Short answer

The net force (F) acting on the particle is proportional to time (t), as found by differentiating the given momentum expression: \(F = \frac{dP}{dt} = 2bt\). Therefore, the correct answer is (A) Proportional to t.

Step-by-step solution

1) Differentiate the Momentum Expression

First, we need to find the time derivative of the momentum expression P = a + b*t^2. To do this, we will apply the power rule of differentiation for each term concerning time: \(\frac{dP}{dt} = \frac{d}{dt}(a + bt^2)\)

2) Differentiate the Constants

Since 'a' and 'b' are constants, the derivative of the first term with respect to time is zero: \(\frac{d}{dt}(a) = 0\)

3) Differentiate the Second Term

Now, we need to differentiate the second term concerning time: \(\frac{d}{dt}(bt^2)= b\frac{d}{dt}(t^2) = 2bt\)

4) Combine the Results

Now, let's combine our results from steps 2 and 3: \(\frac{dP}{dt}= 0 + 2bt = 2bt\)

5) Identify the Net Force

We know that the net force acting on the particle (F) is equal to the time derivative of momentum: \(F =\frac{dP}{dt}\) So, substituting our result for the derivative: \(F = 2bt\)

6) Final Analysis

Now, we can see that the net force (F) acting on the particle is proportional to time (t) since it is directly proportional to 't' multiplied by a constant '2b'. Consequently, the correct option is: (A) Proportional to t

Key concepts

Differentiation

Differentiation is a key concept in calculus, used to find the rate at which a quantity changes. In this scenario, we differentiate the momentum expression with respect to time. The momentum, an essential physics concept, is described by the formula:

• \( P = a + bt^2 \)

Where \(a\) and \(b\) are constants, and \(t\) represents time. To find the net force from this momentum expression, we need to compute the time derivative. By using the power rule of differentiation, which states that the derivative of \(t^n\) is \(nt^{n-1}\), we find how the momentum changes as time progresses.
Applying this rule to \(bt^2\), we derive \(2bt\). The term \(a\) vanishes because it is a constant. Thus, the derived expression \(2bt\) represents the rate of change of momentum with respect to time.

Net Force

The net force is a fundamental concept in physics, representing the overall force acting on an object. When considering linear momentum, the net force is directly linked to the time rate of change of that momentum. In mathematical terms, this is expressed by Newton's second law:

• \( F = \frac{dP}{dt} \)

Upon differentiating the given momentum function \( (P = a + bt^2) \), we discover the net force by substituting the derivative:

• \( F = 2bt \)

This result indicates that the net force is not constant but instead changes with time. Here, \(2b\) is a coefficient that stays constant, while \(t\) denotes time. Consequently, the force acting on the particle varies as time progresses, being directly proportional to \(t\). This insight is critical as it helps predict and analyze the particle's motion.

Proportionality

Proportionality is a concept used frequently in mathematics and physics to describe how one quantity changes in relation to another. When we say a quantity is proportional to another, we imply that there is a constant ratio between the two. In the case of the net force derived from the momentum expression, we concluded:

• \( F = 2bt \)

Here, the force \( F \) is proportional to time \( t \), dictated by the constant \( 2b \). This means that as time increases, the force increases at a rate consistent with the value of \( b \).
To contextualize this in a real-world scenario, think of how a car accelerates: the force applied grows with time as the driver presses the accelerator pedal. Understanding this relationship aids in predicting how the force and subsequent motion of an object will change over time. This foundational principle allows us to foresee the behavior of dynamic systems based on their governing equations.