A child applies a force F⃗ parallel to the x -axis to a a 10.0 -kg sled moving on the frozen sur

step 1 Here in this given problem, mass of the sled moving on the frozen surface that is given as m is

A child applies a force F⃗ parallel to the x -axis to a a...

A child applies a force $\vec{F}$ parallel to the $x$ -axis to a a 10.0 -kg sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the $x$ -component of the force she applies varies with the $x$ -coordinate of the sled as shown in Fig. E6.34. Calculate the work done by the force $\vec{F}$ when the sled moves (a) from $x=0$ to $x=8.0 \mathrm{m} ;$ (b) from $x=8.0 \mathrm{m}$ to $x=12.0 \mathrm{m},(\mathrm{c})$ from $x=0$ to 12.0 $\mathrm{m}$.

A child applies a force $\vec{F}$ parallel to the $x$ -axis to a a 10.0 -kg sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the $x$ -component of the force she applies varies with the $x$ -coordinate of the sled as shown in Fig. E6.34. Calculate the work done by the force $\vec{F}$ when the sled moves (a) from $x=0$ to $x=8.0 \mathrm{m} ;$ (b) from
$x=8.0 \mathrm{m}$ to $x=12.0 \mathrm{m},(\mathrm{c})$ from $x=0$ to 12.0 $\mathrm{m}$.

Key Concepts

Force-Displacement Graph Interpretation

When a force is plotted as a function of displacement, the work done by the force on the object is represented by the area under the curve between two points. This graphical interpretation provides a visual means of understanding how the work accumulates as the object moves and is particularly useful when the force is not constant.

Integration in Work Calculations

Integration is a mathematical tool used to sum up infinitesimally small contributions of work when a force is variable. When a force depends on position, the total work done is obtained by computing the definite integral of the force function with respect to displacement, effectively summing the work done over each differential element of the displacement.

Work

Work in physics is defined as the product of a force acting on an object and the displacement of the object in the direction of that force. When the force is constant, work can be computed simply as the product of force and displacement. However, if the force varies along the displacement, the work done is calculated by integrating the force over the path of the displacement.

Variable Force

A variable force is one whose magnitude or direction changes with position or time, rather than remaining constant. In problems involving variable force, the work done cannot be calculated simply by multiplying force by displacement; instead, the variable nature of the force must be accommodated by integrating the force over the distance through which it acts.

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