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
V = sq
To find the velocity of the crate at the bottom of the ramp, we can use the work-kinetic energy theorem. The work-kinetic energy theorem states that the work done on an object is equal to the change in its kinetic energy.
First, let's find the work done on the crate as it slides down the ramp. The work done is equal to the force applied multiplied by the distance over which the force is applied. In this case, the force is the component of the weight of the crate that acts along the direction of the ramp.
The weight of the crate can be found using the formula:
weight = mass * gravitational acceleration
weight = 31.0 kg * 9.8 m/s^2 = 303.8 N
The component of the weight along the ramp can be found using trigonometry:
force along ramp = weight * sin(angle)
force along ramp = 303.8 N * sin(14 degrees) = 83.1 N
Now we can find the work done:
work = force * distance
work = 83.1 N * 2.6 m = 215.4 J (Joules)
According to the work-kinetic energy theorem, this work done is equal to the change in kinetic energy of the crate.
The initial kinetic energy (K1) of the crate is zero since it is initially at rest.
The final kinetic energy (K2) of the crate can be found using the formula:
kinetic energy = 0.5 * mass * velocity^2
We can rearrange the formula to solve for velocity:
velocity = sqrt(2 * kinetic energy / mass)
Substituting the values we have:
velocity = sqrt(2 * 215.4 J / 31.0 kg) = 7.31 m/s (rounded to two decimal places)
Therefore, the velocity of the crate at the bottom of the ramp is 7.31 m/s (rounded to two decimal places).