Sample Problem 3/3 (Rectilinear Motion)
Sample Problem 3/3 (Rectilinear Motion)
The 250-lb concrete block A is released from
rest in the position shown and pulls the 400-lb
log up the 30° ramp. Plot the velocity of the
block as it hits the ground at B as a function of
the coefficient of kinetic friction μk between the
log and the ramp. Let μk vary between 0 and 1.
Why does the computer not plot results for the
entire range specified?
Problem Formulation
The constant length of the cable is L = 2sC + sA
(see figure). Differentiating this expression
twice yields a relation between the acceleration
of A and C (note that aC = aLOG).
0 = 2a C + a A
(1)
From the free-body diagram for the log
[ΣF
y
= ma y = 0
[ΣFx = ma x ]
]
N − 400 cos(30) = 0
μ k N − 2T + 400 sin(30) =
400
aC
32.2
Substituting N yields,
400μ k cos(30) − 2T + 400 sin(30) =
400
aC
32.2
(2)
From the free-body diagram for block A
[↓ ΣF = ma]
250 − T =
250
aA
32.2
(3)
Maple will be used to solve the three equations above for aA, aC and T in terms of
μk. Since the accelerations are constant, v A2 = 2a A d where d is the vertical
distance through which block A has fallen. Thus, the velocity of A when it strikes
the ground (d = 20 ft) is
v Af = 40a A
Maple will automatically substitute aA yielding the required expression relating
vAf and μk.
Maple Worksheet
> restart; Digits:=5:
> eqn1:= 0=2*aC+aA;
eqn1 := 0 = 2 aC + aA
> eqn2:= 400*mu[k]*cos(theta)-2*T+400*sin(theta)=400/32.2*aC;
eqn2 := 400 μk cos ( θ ) − 2 T + 400 sin( θ ) = 12.422 aC
> eqn3:= 250-T=250/32.2*aA;
eqn3 := 250 − T = 7.7640aA
> theta:=30*Pi/180:
> soln:=solve({eqn1,eqn2,eqn3},{aA,aC,T});
soln := { aA = −15.935 μk + 13.800, aC = 7.9675 μk − 6.9000, T = 142.86 + 123.72 μk }
Note that the accelerations may be either positive or negative depending on the
value of μk . The largest value of μk for which the block will move up can thus
be found by solving the equation aA=0 for μk . This yields μk = 13.8/15.935 =
0.866.
> assign(soln);
> vAf:=sqrt(2*aA*20);
vAf := −637.40μk + 552.00
> plot(vAf,mu[k]=0..1, labels=["mu_k"," "], title="vB (ft/sec)");
Note that results are not plotted beyond the limiting value for μk (0.866) that
was determined above. From a numerical point of view this occurs because
Maple will not plot imaginary answers. Whenever imaginary or complex values
result there is usually some physical explanation. In this problem, the physical
explanation is that the log will not slide up the incline if the coefficient of friction
is too large.
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