Starting from rest, a 11.70 kg suitcase slides 3.14 m down a frictionless ramp inclined at 40° from the floor. The suitcase then slides an additional 4.68 m along the floor before coming to a stop.
(a) Determine the suitcase's speed at the bottom of the ramp.
(b) Determine the coefficient of kinetic friction between the suitcase and the floor.
(c) Determine the change in mechanical energy due to friction.
1)v=sqrt(2*9.8*3.14sin40)=6.29m/s (2)umgd=1/2mv^2 u*9.8*4.68=0.5(6.29)^2 u=0.43 (3)change in mechanical energy=workdone by friction=0.43*11.7*9.8*46.8=2307.4J
To solve this problem, we can use the principles of conservation of energy and the work-energy theorem. Here's how we can approach each part of the problem:
(a) To determine the suitcase's speed at the bottom of the ramp, we can use the principle of conservation of mechanical energy. The initial mechanical energy at the top of the ramp is equal to the final mechanical energy at the bottom of the ramp.
The initial mechanical energy is given by the potential energy due to the height of the suitcase above the bottom of the ramp: U_i = mgh_i, where m is the mass of the suitcase, g is the acceleration due to gravity, and h_i is the vertical height of the suitcase at the top of the ramp.
The final mechanical energy is given by the kinetic energy of the suitcase at the bottom of the ramp: K_f = (1/2)mv_f^2, where v_f is the velocity of the suitcase at the bottom of the ramp.
Since there is no friction on the ramp, the change in mechanical energy is 0 (neglecting any losses due to air resistance). Therefore, we can set the initial mechanical energy equal to the final mechanical energy:
mgh_i = (1/2)mv_f^2
We can then solve for v_f:
v_f = sqrt(2gh_i)
Substituting the given values, we have:
m = 11.70 kg
g = 9.8 m/s^2
h_i = 3.14 m
v_f = sqrt(2 * 9.8 m/s^2 * 3.14 m) = 8.86 m/s
So the suitcase's speed at the bottom of the ramp is 8.86 m/s.
(b) To determine the coefficient of kinetic friction between the suitcase and the floor, we can use the work-energy theorem. The work done by friction is given by W_friction = force_friction * distance, where force_friction is the frictional force between the suitcase and the floor, and distance is the distance the suitcase slides along the floor.
The work done by friction is equal to the change in mechanical energy, which is 0 (since the suitcase comes to a stop). Therefore:
W_friction = 0
Since the force_friction is opposite to the motion, the work done by friction is negative. Therefore:
force_friction * distance = 0
We can rearrange this equation to solve for the force_friction:
force_friction = 0 / distance = 0 N
Since the force of friction is 0, this means that the coefficient of kinetic friction between the suitcase and the floor is also 0. In other words, there is no friction between the suitcase and the floor.
(c) Since there is no friction between the suitcase and the floor, there is no change in mechanical energy due to friction. Therefore, the change in mechanical energy is 0 J.