A 31.0 kg crate, initially at rest, slides down a ramp 2.6 m long and inlined 14 degrees with the horizontal. Using the work-kinetic energy theorem and disregarding friction, find the velocity of th crate at the bottom of the ramp.
4.407
To find the velocity of the crate at the bottom of the ramp using the work-kinetic energy theorem, we need to calculate the work done on the crate and equate it to the change in kinetic energy.
The work-kinetic energy theorem states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, it can be represented as:
Work = Change in Kinetic Energy
In this case, since the crate is initially at rest, the initial kinetic energy is zero. Therefore, the equation becomes:
Work = Final Kinetic Energy
The work done on an object is equal to the force applied to it multiplied by the distance over which the force is applied. In this case, the force that is doing the work is the component of the weight of the crate parallel to the ramp. To calculate this force, we can use the following formula:
Force = Mass * Acceleration
Since the crate is sliding down the ramp, the acceleration is equal to the acceleration due to gravity multiplied by the sine of the angle of inclination:
Acceleration = g * sin(theta)
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and theta is the angle of inclination (14 degrees).
Now that we have the force, we can calculate the work done:
Work = Force * Distance
Distance in this case is the length of the ramp, which is given as 2.6 m.
Next, we equate the work done to the change in kinetic energy:
Work = Final Kinetic Energy
Finally, we can solve for the final kinetic energy, which is given as:
Final Kinetic Energy = (1/2) * Mass * Velocity^2
Using these equations, we can calculate the velocity of the crate at the bottom of the ramp. Let me calculate it for you.