Question

A block is placed on a rough horizontal plane. A time dependent horizontal force $F = kt$ acts on the block, where $k$ is a positive constant. The acceleration - time graph of the block is:

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is B

To solve the problem of determining the acceleration-time graph for a block subjected to a time-dependent horizontal force on a rough horizontal plane, we need to consider a few physics concepts:

Force and Acceleration Relationship: According to Newton's second law of motion, the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, this is expressed as:

\(F = ma\)

Where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration.

Given Force Expression: The force acting on the block here is given as \(F = kt\), where \(k\) is a constant and \(t\) is the time.

Acceleration Calculation: Using Newton's second law, substitute the force into the equation:

\(ma = kt\)

Solving for acceleration \(a\), we get:

\(a = \frac{kt}{m}\)

Graph Characteristics: The expression \(a = \frac{kt}{m}\) shows that acceleration \(a\) is directly proportional to time \(t\). This indicates a linear relationship between acceleration and time. Thus, as time increases, acceleration increases linearly.

Selecting the Correct Graph: Among the provided options, we need a graph that shows acceleration increasing linearly with time, which corresponds to a straight line with a positive slope.

The correct graph is:

The correct option is a graph that displays a linear increase in acceleration with respect to time, confirming the proportionality indicated in the derived equation \(a = \frac{kt}{m}\).