A 20. kg block is placed at the top of a 10. meter-long inclined plane. The block starts from rest and slides without friction down the length of the incline. (A diagram shows a height of 5.0 m, the 10. m length of incline/hypotenuse and the 20. kg block sliding on the hypotenuse.)
1) find the gravitational potential energy of the block at the TOP of the incline.
2) Find the kinetic energy of the block just as it reaches the bottom of the incline.
3) Sketch a graph of the GPE of the block as a function of its KE for the complete slide.
I need help setting up #1 and #2.
For #3, I know KE goes up as GPE goes down and vice versa. How do I show this on a graph?
Thank you for your help.
To solve questions 1 and 2, you will need to use the formulas for gravitational potential energy (GPE) and kinetic energy (KE). Here's how to set up the calculations:
1) Gravitational Potential Energy at the top of the incline:
The formula for gravitational potential energy is given by:
GPE = m * g * h
where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth), and h is the height.
Given:
m = 20 kg (mass of the block)
h = 5.0 m (height at the top of the incline)
Substituting these values into the formula, calculate GPE:
GPE = 20 kg * 9.8 m/s^2 * 5.0 m
2) Kinetic Energy at the bottom of the incline:
The formula for kinetic energy is given by:
KE = 0.5 * m * v^2
where m is the mass of the object and v is the velocity.
Since the block slide without friction, the conservation of energy tells us that the GPE at the top of the incline (question 1) is converted entirely into kinetic energy at the bottom of the incline (question 2). So, the GPE at the top of the incline can be equated to the KE at the bottom:
m * g * h = 0.5 * m * v^2
Given:
m = 20 kg (mass of the block)
g = 9.8 m/s^2 (acceleration due to gravity)
h = 5.0 m (height at the top of the incline)
Solve the equation for v to find the velocity at the bottom of the incline.
Now, for question 3:
You are correct that GPE and KE are inversely related. As GPE decreases, KE increases, and vice versa. To show this on a graph, you can plot GPE on the y-axis and KE on the x-axis.
Start the graph with GPE at its maximum value (at the top of the incline) and KE at its minimum value (zero at the top of the incline). As the block slides down the incline, GPE decreases while KE increases. The graph would show a downward-sloping line for GPE and an upward-sloping line for KE.
Note that the total mechanical energy (sum of GPE and KE) remains constant throughout the slide, representing the conservation of energy.
I hope this helps! Let me know if you have any more questions.