Posted by Chad on Thursday, August 9, 2012 at 10:48am.
Box, mass m 1 , at rest against a compressed spring, spring constant k and compression of Δx, at the top of a
ramp, height h, is released. It slides down the ramp, angle θ, with friction, coefficient μ. Half way down the
ramp, it bumps into and sticks to another box, mass m 2 , and together, they slide to the bottom. If m 1 = 2.0 kg,m2= 3.0 kg, k = 500 N/m, Δx = 0.40 m, h = 5.0 m, μ= 0.30, θ= 30.0O, how long does it take both boxes to reach the bottom (starting from when the first box is released)? The spring is parallel to the ramp surface and the Δx is measured parallel to the ramp surface.

Physics  Elena, Thursday, August 9, 2012 at 3:53pm
k •Δx²/2=m1•vₒ²/2, ….. (1)
vₒ = Δx•sqrt(k/m1) =0.4•sqrt(500/2) =6.32 m/s
The distance from start to the bottom is
s=h/sinθ = 5/0.5 = 10 m.
Two parts s1 = s2 =10/2 =5 m.
The law of conservation of energy for s1:
m1•vₒ²/2 +ΔPEW(fr) = m1•v1²/2,
Taking into account (1) we obtain
k •Δx²/2 +m1•g•h/2  μ•m1•g•cosθ•h/2•sinθ = m1•v1²/2
v1= sqrt{ k •Δx²/m1 +g•h[1 (μ/tanθ)] }=
=sqrt{500•0.16/2 +9.8•5•[1(0.3/tan30º]} = 7.97 m/s.
From kinematics:
a1=(v1²vₒ²)/2s = (7.97²  6.32²)/2•5 = 2.358 m/s²
v1=vₒ +a•t1,
t1= (v1vₒ)/a=(7.976.32)/2.358 =0.6997 = 0.7 s.
The law of conservation of linear momentum
m1•v1+0 =(m1+m2) •u,
u =m1•v1/(m1+m2) = 2•7.97/5=3.2 m/s.
The law of conservation of energy for s2:
(m1+m2)•u²/2 +ΔPEW(fr) = m1•u1²/2,
(m1+m2)•u²/2 + (m1+m2)•g•h/2  μ•(m1+m2)•g•cosθ•h/2•sinθ = (m1+m2)•u1²/2,
u1=sqrt{u²+g•h[1(μ/tanθ)]} =
=sqrt{3.2² +9.8•5(10.3/tan30º)} =5.81 m/s,
From kinematics:
a=(u1²u²)/2•s=(5.81²3.2²)/2•5 = 2.352 m/s²
u1=u+a•t2
t2=(u1u)/a=(5.813.2)/2.352 =1.1 s.
t= t1+t2 =0.7+1.1 =1.8 s.
Answer This Question
Related Questions
 Physics  A spring of spring constant k is attached to a support at the bottom ...
 Physics  A box slides from rest down a frictionless ramp inclined at 35.0° with...
 physics  A dude is spinning a 1.0 kg mass on the end of a 1.5 meter string. He ...
 physics  A 10 kg block slides 4.0 m down a ramp (having a coefficient of ...
 Physics  A block of mass m = 2.4 kg is released from rest at a height of H = 3....
 Physics  A child's game consists of a block that attaches to a table with a ...
 Physics  A(n) 160 g block is pushed by an external force against a spring with ...
 physics  A block of mass m = 2.7 kg is released from rest at a height of H = 0....
 Physics  A block of mass m = 5.2 kg is released from rest at a height of H = 4....
 physics  The 2.kg mass is heading toward a rampthere’s no friction. Notice ...
More Related Questions