A 20g particle is moving to the left at 30 m/s. How much net work must be done on the particle to cause it to move to the right at 30 m/s.
The answer is 0J of work, and i understand the method of equations, but i don't understand the logic behind it.
Thanks for any help.
You are not changing the kinetic energy and you do not have to change the potential energy to change direction. Therefore no work is required, if the reversal is done in a frictionless manner.
You could let its direction be reversed by a spring, or by a U-shaped skateboard ramp, for example.
To understand the logic behind why the net work required is 0J, we need to consider the concept of work and its relationship with velocity.
Work is defined as the transfer of energy that occurs when a force is applied over a displacement. Mathematically, work (W) is calculated as the product of the applied force (F) and the displacement (d) in the direction of the force, multiplied by the cosine of the angle (θ) between the force and displacement vectors:
W = F * d * cos(θ)
In this case, the particle is initially moving to the left at 30 m/s. To change its direction of motion to the right, we need to apply a force in the opposite direction.
When we calculate the work done on an object, the direction of the force and the displacement are important. If the force and displacement are in the same direction (θ = 0°), the work done is positive. If the force and displacement are in opposite directions (θ = 180°), the work done is negative.
In this scenario, the net work required to change the particle's motion from left to right is 0J. This is because the force required to make this change is equal in magnitude and opposite in direction to the force that was initially applied causing the leftward motion.
Therefore, the work done in the first direction is canceled out by the work done in the opposite direction, resulting in a net work of 0J.
To understand the logic behind why no net work is required to cause the particle to move from left to right at the same speed, let's break down the concept of work and its relation to the particle's velocity.
Work is defined as the transfer of energy resulting from the application of a force over a distance. Mathematically, the work done on an object is given by the equation:
Work = Force * Distance * cos(θ)
In this case, the force required to move the particle in the opposite direction is equal in magnitude but opposite in direction to the force that is slowing it down. As a result, the angle θ between the applied force and the displacement is 180 degrees, and the cos(180) = -1.
Now, let's consider the equation in the scenario where the particle is moving to the left at 30 m/s. The negative velocity indicates that the particle is encountering resistance or a force in the opposite direction that is causing it to decelerate. If no net force is applied, the particle will eventually come to a stop.
To make the particle move to the right at 30 m/s while experiencing the same magnitude of force, no additional work needs to be done. This is because the force applied will simply counteract the decelerating force, allowing the particle to maintain a constant velocity. Mathematically, this can be represented as:
Work = Force * Distance * cos(180) = Force * Distance * (-1) = -Force * Distance
Since the work done is the product of force and distance, and the distance is the same for both scenarios, if we multiply the force by -1, the work done becomes negative. Therefore, the net work required to cause the particle to move from left to right at the same speed is 0J (joules), signifying that no additional work is needed.
In summary, because the force required to move the particle in the opposite direction is equal and opposite to the force that is slowing it down, no additional net work is necessary to cause the particle to move from left to right at the same speed.