Answer:
76.3 J
Explanation:
I'm assuming the distance of 4.60 m is along the incline, not the vertical distance from the bottom. I'll call this distance d, so h = d sin θ.
Initial energy = final energy
Energy in spring = gravitational energy + kinetic energy + work by friction
E = mgh + 1/2 mv² + Fd
We need to find the force of friction. To do that, draw a free body diagram.
Normal to the incline, we have the normal force pointing up and the normal component of weight (mg cos θ).
Sum of the forces in the normal direction:
∑F = ma
N - mg cos θ = 0
N = mg cos θ
Friction is defined as:
F = Nμ
Plugging in the expression for N:
F = mgμ cos θ
Substituting:
E = mgh + 1/2 mv² + (mgμ cos θ) d
E = mg (d sin θ) + 1/2 mv² + (mgμ cos θ) d
E = mgd (sin θ + μ cos θ) + 1/2 mv²
Given:
m = 1.45 kg
g = 9.90 m/s²
d = 4.60 m
θ = 29.0°
μ = 0.45
v = 5.10 m/s
Solving:
E = mgd (sin θ + μ cos θ) + 1/2 mv²
E = (1.45) (9.80) (4.60) (sin 29.0 + 0.45 cos 29.0) + 1/2 (1.45) (5.10)²
E = 76.3 J
