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### Pushing a mass up a slope and energy

Please note: knowledge of the SUVAT equations, friction and trigonometry is required for this thread.

Energy comes in many forms, and here we will focus on kinetic energy (KE) and potential energy (PE). Below are the equations of KE and PE.

change in KE = 0.5*m*[v1^2 - v2^2]

change in PE = m*g*[h2 - h1]

where m = mass

v1 = beginning velocity of mass

v2 = final velocity of mass

g = gravitational acceleration = 9.81 m/s^2

h1 = beginning height of mass

h2 = final height of mass

Here’s the scenario: a box is on a slope and is at rest (a velocity of 0 m/s). A man comes and pushes the box with constant force up the slope.

Let’s consider the energy of the box. Because the velocity of the box was originally 0 m/s and the man is now pushing it, which means it has some velocity because it is moving, the box’s KE has increased. Because the distance of the box from the ground has increased (the box is being pushed up the slope) the box’s PE has increased.

To find the change in KE, we will need to find the value of the beginning velocity, v1, and the value of the final velocity, v2, This can be done using the SUVAT equations and F=m*a, and I will be making a thread on these equations soon. In examples the mass will be constant and given, therefore we have everything we need to find the change in KE

To calculate PE we need the change in vertical distance of the box. This is complicated by the fact that the box is not moving in the vertical direction, but moving up a slope. Let us construct a right-angled triangle: one line being the ground (ground line), the line perpendicular (at 90 degrees) to the ground line being the change in vertical distance (vertical line) and the hypotenuse (longest line of the triangle) being the distance travelled by the mass (distance line). The angle between the slope and the ground will be the same angle the distance line makes with the ground line in our newly drawn triangle. We can calculate the distance travelled by the mass using the SUVAT equations, and thus find the length of the distance line. We can know complete our drawn triangle using trigonometry to find the length of the vertical line. The length of the vertical line is equal to the change in vertical height of the mass. Thus, we now have everything we need to find PE.

Some systems will ignore friction, perhaps because it is very small. Some systems will include friction because it is not small. Let’s say that friction is not a small term. We already know how to calculate the friction force from its equation: friction = coefficient of friction * reaction force. We also know that the work done (energy) of a body is: work done = force * distance. Substituting in the friction force to find the work done by the friction force we have: work done by friction force = coefficient of friction * reaction force * distance travelled. Thus we can now factor this into the system.

Remember the law of conservation of energy: energy cannot be created or destroyed, but can be transferred from one form (e.g. PE) to another (e.g. KE). Therefore, the changes in KE, PE and work done by friction must add up to be zero.

3 years ago

Answered by Daniel, an A Level Maths tutor with MyTutor

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