A 1.40 -kg object slides to the right on a surface having a coefficient of kinet
ID: 2212624 • Letter: A
Question
A1.40-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Figure a). The object has a speed ofvi=3.10m/swhen it makes contact with a light spring (Figure b) that has a force constant of50.0 N/m.The object comes to rest after the spring has been compressed a distanced(Figure c). The object is then forced toward the left by the spring (Figure d) and continues to move in that direction beyond the spring's unstretched position. Finally, the object comes to rest a distanceDto the left of the unstretched spring (Figure e).
(a) Find the distance of compressiond.
m
(b) Find the speedvat the unstretched position when the object is moving to the left (Figure d).
m/s
(c) Find the distanceDwhere the object comes to rest.
m
.
Explanation / Answer
Use the law of conservation of energy
Kinetic energy of mass = work absorbed by spring + energy to overcome friciton
KE = Ws + Wf
(1/2)(mV^2) = (1/2)kd^2 + (mu)(m)(g)d
where
m = mass = 1.40 kg (given)
V = velocity of mass = 3.10 m/sec. (given)
k = spring constant = 50 N/m (given)
d = distance at which spring was compressed
mu = coefficient of friction = 0.250 (given)
g = acceleration due to gravity = 9.8 m/sec^2 (constant)
Substituting values,
(1/2)(1.40)(3.10)^2 = (1/2)(50)d^2 + (0.250)(1.30)(9.8)(d)
6.727 = 25d^2 + 3.185(d)
Rearranging the above,
25d^2 + 3.185d - 6.727 = 0
Using the quadratic equation,
solve d in mts
<< Find the speed v at the unstretched position when the object is moving to the left. >>
Use the law of conservation of energy again, i.e.,
Energy from spring = Kinetic energy of mass + Work to overcome friction
Ws = KE + Wf
(1/2)(50)(d)^2 = (1/2)(1.30)V^2 + (0.25)(1.30)(9.8)(d)
<< Find the distance D where the object comes to rest. >>
Working formula is
Vf^2 - V^2 = 2aD
where
Vf = final velocity = 0 (when object comes to rest)
a = acceleration
D = distance when mass will stop
From Newton's 2nd Law of Motion, F = ma
where
f = frictional force = (mu)(mg)
Therefore,
(mu)(mg) = ma
subs a value in above eqn
the negative sign attached to the acceleration. This simply means that the mass was slowing down as it was moving away from the spring.
Solve for "D"
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