Suppose the sled in Exercise 6.34 is initially at rest at...

Suppose the sled in Exercise 6.34 is initially at rest at $x=0 .$ Use the work-energy theorem to find the speed of the sled at (a) $x=8.0 \mathrm{m}$ and $(\mathrm{b}) x=12.0 \mathrm{m}$ . You can ignore friction between the sled and the surface of the pond.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, which is calculated using the formula (1/2)·m·v². This concept is key in dynamics and energy analysis because it quantifies how the motion of an object changes as work is done on it, providing insight into its speed and state of motion.

Work-Energy Theorem

The work-energy theorem is a fundamental principle in mechanics which states that the net work done on an object is equal to its change in kinetic energy. It connects the forces acting on the object with its motion by showing that any work done on the object results in a change in its speed or kinetic energy.

Net Work

Net work is the total work done by all forces acting on an object along its path of movement. It is a crucial concept in applying the work-energy theorem, as it represents the cumulative effect of all forces on the kinetic energy of an object, allowing for the determination of changes in speed.

Transcript

00:01 In the given problem, mass of the sledge is given as m is equal to 10 .0 kilogram.

00:15 Now, the graphical representation of the variation of force and displacement is given as here.

00:27 The force, first of all, varies from zero to the maximum value of 10 newton, the force along.

00:34 The component of the component of the force along x -axis and then it drops to 0 here this maximum force is 10 newton if this is f in newton and here this distance is up to which the distance up to which the force is increased that is 0 then this is 4 then this is 8 if the distance is measured in meter and then this force drops to zero at a distance of 12 meter now in the first part of the problem using work energy theorem we have to find the speed achieved by this sledge at a distance of eight meter so first of all we will find work done on the sledge from x equals to zero to x equals to 8 meter and that will be given by area of triangle if we name it o, a, b, and c.

01:51 Then it will be area of triangle o, a, b, and that is given as the half of the product of base ob and perpendicular means height, a, b.

02:05 So here this is is half into ob, which is 8 meter into height, which is 10 newton.

02:15 So this work done comes out to be 40 newton into meter, or we can say this is 40 g's.

02:24 Now using work energy theorem, we know that the energy, the kinetic energy, stored in the sledge, is equal to in the sledge is equal to the work done on it.

02:53 Work done on it.

02:55 So here it becomes half mv square is equal to the total work done in going from 0 to 8 meter.

03:04 So this is half into mass which is given as 10 kilogram.

03:08 V square here this is missing is equal to w means the work done which we just found as 40 jules so here it comes out to be 5 so v square is equal to 40 by 5 which comes out to be 8 hence we conclude that the speed of the sledge at the end of the distance 8 meter is given as a square root of 8 so it comes out to be 2 .83 meter per second and this is the answer for the first part of the problem...

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