A light spring of force constant 3.55 N/m is compressed by 8.00 cm and held betw

Question

A light spring of force constant 3.55 N/m is compressed by 8.00 cm and held between a 0.250 kg block on the left and a 0.600 kg block on the right. Both blocks are at rest on a horizontal surface. The blocks are released simultaneously so that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction between each block and the surface is the following. In each case, assume that the coefficient of static friction is greater than the coefficient of kinetic friction. Let the positive direction point to the right.

Uk .250 kg block .600 kg block

0.00 ___________m/s _____________m/s

0.10 ___________m/s _____________m/s

0.40 ___________m/s _____________m/s

Explanation / Answer

|Fs| = kx = 3.55 * 0.08 = 0.284 N

part a )

when no friction , use energy conservation

(K+U)i = (K+U)f

1/2kx^2 = 1/2*m1*v1f^2 + 1/2*m2*v2f^2

1/2 * 3.55 * (0.08^2) = 1/2 * 0.250 * v1f^2 + 1/2*0.6 * v2f^2

conservation of linear momentum

m1v1i + m2v2i = m1v1f + m2v2f

0 = 0.250 v1f(-x) + 0.600 x v2f(+x)

v1f = 2.4*v2f

0.01136 = 1/2*0.250 * (2.4v2f)^2 + 1/2 * 0.600 * v2f^2

0.01136 = 0.72 v2f^2 + 0.3 *v2f^2

0.01136 = 1.02 v2f^2

v2f = 0.1055 m/s ( 0.6 kg )

v1f = 0.253 m/s (0.250 kg)

part b )

for lighter block

sum of forces in y direction = m*ay

n - 0.250 * 9.8 = 0

n = 2.45 N

f = mu*n = 0.1 * 2.45 = 0.245 N

|Fs| = 0.284

here spring force is larger than friction force so block will move

for heavier block

n = 0.6 * 9.8 = 5.88

f = 0.1*5.88 = 0.588 N

for heavier block friction force is larger than spring force , so heavier block wont move .

|Fs| = kx

0.245 = 3.55 *xf

xf = 0.069 m

Ki + Ui - f*d = Kf + Uf

0 + 0.01136 - [0.245 * ( 0.08 - 0.069) = 1/2 * 0.250 *v1f^2 + 1/2 * 3.55 * (0.069)^2

8.665 x 10^-3 = 0.125 v1f^2 + 8.450775 x 10^-3

v1f = 0.0414 m/s

v2f = 0

part c )

f for lighter block = 0.4 * 2.45 = 0.98 N

The force of static friction must be at least as large. the 0.284 N spring force is small to produce motion of either block .

each has 0 zero velocity

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