1) a loaded cart with a mass of 96 kg sits at reat atop a 40 degree inclined plane that measures 20.5m long. accidentally the cart starts so to roll down the inclined plane with a frictional force of 5550N working against it for the intire 20.5m. deteremine the speed at the bottom og the incline plane.
2) a 1,420kg boulder at rest, begins to roll down a 52m long hillside. while rolling, the ground causes a frictional force of 12,040N. the boulder arrives at the bottom of the hillside with a speed of 19.5m/s. determine the boulder's original height.
1. Wc = 96kg * 9.8N/kg = 940.8N @ 40deg = Weight of cart.
Fp = 940.8sin40 = 604.7N = Force parallel to the plane.
Fv = 940.8cos40 = 720.7n = Force perpendicular to the plane.
Ff = 555N = Force due to friction.
Fn = Fp - Ff = 604.7 - 555 = 49.7N =
Net force.
a = Fn/m = 49.7 / 96 = 0.518m/s^2.
Vf^2 = Vo^2 + 2ad
Vf^2 = 0 + 2*0.518*21.5 = 22.27,
Vf=4.7m/s = Speed at bottom of incline.
1) To determine the speed of the cart at the bottom of the inclined plane, we can use the principle of conservation of energy. The initial potential energy of the cart at the top of the incline is given by:
Potential Energy = m * g * h
where m is the mass of the cart, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the inclined plane. On the inclined plane, the cart's potential energy is converted into kinetic energy:
Kinetic Energy = (1/2) * m * v^2
where v is the speed of the cart at the bottom of the incline.
The work done against friction is given by:
Work = frictional force * distance
In this case, the frictional force is given as 5550 N and the distance is 20.5 m.
Since the work done against friction is equal to the energy lost due to friction, we can equate the two expressions:
Work = m * g * h - (1/2) * m * v^2
Substituting the given values, we have:
5550 N * 20.5 m = 96 kg * 9.8 m/s^2 * h - (1/2) * 96 kg * v^2
Now, let's solve for v:
v^2 = (2 * 96 kg * 9.8 m/s^2 * h - 5550 N * 20.5 m) / 96 kg
v = sqrt((2 * 96 kg * 9.8 m/s^2 * h - 5550 N * 20.5 m) / 96 kg)
2) To determine the boulder's original height, we can again use the principle of conservation of energy. The initial potential energy of the boulder at the top of the hillside is given by:
Potential Energy = m * g * h
where m is the mass of the boulder, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the hillside. As the boulder rolls down the hillside, its potential energy is converted into kinetic energy. The work done against friction is given by:
Work = frictional force * distance
In this case, the frictional force is given as 12,040 N and the distance is 52 m.
Since the work done against friction is equal to the energy lost due to friction, we can equate the two expressions:
Work = m * g * h - (1/2) * m * v^2
Substituting the given values, we have:
12040 N * 52 m = 1420 kg * 9.8 m/s^2 * h - (1/2) * 1420 kg * (19.5 m/s)^2
Now, let's solve for h:
h = (12040 N * 52 m + (1/2) * 1420 kg * (19.5 m/s)^2) / (1420 kg * 9.8 m/s^2)
h = (627,680 N*m + 132,467.5 N*m) / 13916 N
h = 759,147.5 N*m / 13916 N
h ≈ 54.57 m
Therefore, the boulder's original height is approximately 54.57 meters.