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 the crate at the bottom of the ramp
If there is no friction, the angle of the slope doesnt matter. The mass won't matter either, after you set
M g H = (1/2) M V^2
and solve for V.
H is the vertical distance that it falls. Since you know the length L and the angle,
H = L sin 14.
To find the velocity of the crate at the bottom of the ramp using the work-kinetic energy theorem, you need to consider the potential energy and kinetic energy of the system.
The work-kinetic energy theorem states that the net work done on an object is equal to its change in kinetic energy:
Work = Change in Kinetic Energy
First, let's calculate the potential energy of the crate at the top of the ramp. The potential energy (PE) of an object at a height h is given by the formula:
PE = m * g * h
where m is the mass of the crate, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the vertical height.
In this case, the crate is initially at rest and at the top of the ramp, so its potential energy is:
PE = m * g * h = 31.0 kg * 9.8 m/s² * 2.6 m * sin(14°)
Next, let's calculate the kinetic energy (KE) of the crate at the bottom of the ramp. The kinetic energy of an object is given by the formula:
KE = (1/2) * m * v²
where v is the velocity of the object.
Since the crate starts from rest, its initial kinetic energy is zero. At the bottom of the ramp, all the potential energy is converted to kinetic energy. So we equate the potential energy at the top with the kinetic energy at the bottom:
PE = KE
m * g * h = (1/2) * m * v²
Now, we can solve for the velocity (v) of the crate at the bottom of the ramp:
v = sqrt((2 * g * h) / sin(14°))
Substituting the given values:
v = sqrt((2 * 9.8 m/s² * 2.6 m) / sin(14°))
Calculate the value of v:
v ≈ sqrt(50.96 / 0.2419) ≈ sqrt(210.59)
v ≈ 14.52 m/s
Therefore, the velocity of the crate at the bottom of the ramp, disregarding friction, is approximately 14.52 m/s.
To find the velocity of the crate at the bottom of the ramp, we can use the work-kinetic energy theorem, which states that the work done on an object is equal to its change in kinetic energy. In this case, the work done on the crate is due to its weight, and the change in kinetic energy will be equal to the final kinetic energy of the crate.
The work done by the weight of the crate can be calculated using the formula:
W = mgh,
where W is the work done, m is the mass of the crate, g is the acceleration due to gravity, and h is the height of the ramp.
In this case, the vertical height of the ramp can be calculated using the formula:
h = L*sin(theta),
where L is the length of the ramp and theta is the angle of inclination.
Plugging in the given values, we have:
h = 2.6 m * sin(14 degrees)
Next, we can calculate the work done:
W = 31.0 kg * 9.8 m/s^2 * h
Now, the work done is equal to the change in kinetic energy of the crate:
W = (1/2) * m * v^2,
where v is the velocity of the crate at the bottom of the ramp.
Setting the expressions for work equal to each other, we have:
31.0 kg * 9.8 m/s^2 * h = (1/2) * m * v^2
Simplifying the equation, we get:
9.8 m/s^2 * h = (1/2) * v^2
Since we want to find the velocity, we rearrange the equation:
v^2 = 2 * 9.8 m/s^2 * h
v = √(2 * 9.8 m/s^2 * h)
Finally, plug in the value of h calculated earlier and solve for v:
v = √(2 * 9.8 m/s^2 * 2.6 m * sin(14 degrees))
v ≈ 7.79 m/s
Therefore, the velocity of the crate at the bottom of the ramp is approximately 7.79 m/s.