A 15 kg crate is held with a rope at the top of a metal ramp that has an incline of 25(degrees) with the horizontal. The rope breaks and slides down the ramp a distance of 5.50 meters before it hits the floor. If the coefficient of friction between the crate and the metal ramp is 0.27, what is the final velocity of the crate when it reaches the bottom of the ramp?
To find the final velocity of the crate when it reaches the bottom of the ramp, we can use the principle of conservation of mechanical energy.
First, let's break down the problem and the different forces acting on the crate:
1. The force of gravity: This is the weight of the crate, given by the formula: F_gravity = mass * gravity, where mass is the mass of the crate (15 kg) and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = 15 kg * 9.8 m/s^2 = 147 N
2. The normal force: This is the force exerted by the ramp on the crate perpendicular to the surface of the ramp. It is equal in magnitude and opposite in direction to the component of the weight of the crate perpendicular to the ramp.
normal force = F_gravity * cos(angle of incline)
normal force = 147 N * cos(25 degrees) = 132.50 N
3. The force of friction: This is the force opposing the motion of the crate along the ramp. It is given by the formula: F_friction = coefficient of friction * normal force.
F_friction = 0.27 * 132.50 N = 35.775 N (rounding to 3 decimal places)
4. The net force acting on the crate: This is the vector sum of the forces acting on the crate. Since the crate is sliding down the ramp, the net force is determined by the component of the weight of the crate parallel to the ramp minus the force of friction.
net force = F_gravity * sin(angle of incline) - F_friction
net force = 147 N * sin(25 degrees) - 35.775 N = 58.786 N (rounding to 3 decimal places)
Using Newton's second law, F_net = mass * acceleration, we can find the acceleration of the crate:
acceleration = net force / mass
acceleration = 58.786 N / 15 kg = 3.919 m/s^2 (rounding to 3 decimal places)
Now, let's use the kinematic equation to find the final velocity of the crate when it reaches the bottom of the ramp:
v^2 = u^2 + 2as
Here, u is the initial velocity of the crate (0 m/s), a is the acceleration we just calculated (3.919 m/s^2), s is the distance traveled down the ramp before the crate hits the floor (5.50 m), and v is the final velocity we need to find.
Plugging in the values into the equation:
v^2 = 0^2 + 2 * 3.919 m/s^2 * 5.50 m
v^2 = 0 + 42.1097 m^2/s^2 (rounding to 4 decimal places)
Taking the square root of both sides:
v = √(42.1097 m^2/s^2)
v = 6.489 m/s (rounding to 3 decimal places)
Therefore, the final velocity of the crate when it reaches the bottom of the ramp is approximately 6.489 m/s.