Question 11.SP.8: Block C starts from rest and moves down with a constant acce...
Block C starts from rest and moves down with a constant acceleration. Knowing that after block A has moved 1.5 ft its velocity is 0.6 ft/s, deter- mine (a) the acceleration of A and C, (b) the change in velocity and the change in position of block B after 2.5 seconds.
STRATEGY: Since you have blocks connected by cables, this is a dependent-motion problem. You should define coordinates for each mass and write constraint equations for both cables.
MODELING and ANALYSIS: Define position vectors as shown in Fig. 1, where positive is defined to be down.
Constraint Equations. Assuming the cables are inextensible, you can write the lengths in terms of the defined coordinates and then differentiate.
Cable 1: x_{A}+\left(x_{A}-x_{B}\right)=\text { constant }
Differentiating this, you find
2 v_{A}=v_{B} \quad \text { and } \quad 2 a_{A}=a_{B} (1)Cable 2: 2 x_{B}+x_{C}=\text { constant }
Differentiating this, you find
v_{C}=-2 v_{B} \quad \text { and } \quad a_{C}=-2 a_{B} (2)Substituting Eq. (1) into Eq. (2) gives
v_{C}=-4 v_{A} \quad \text { and } \quad a_{C}=-4 a_{A} (3)Motion of A. You can use the constant-acceleration equations for block A:, as
v_{A}^{2}-v_{A_{0}}^{2}=2 a_{A}\left[x_{A}-\left(x_{A}\right)_{0}\right] \quad \text { or } \quad a_{A}=\frac{v_{A}^{2}-\left(v_{A}\right)_{0}^{2}}{2\left[x_{A}-\left(x_{A}\right)_{0}\right]} (4)a. Acceleration of A and C. You know v_C \text{and} a_C are down, so from Eq. (3), you also know v_{A} and a_{A} are up. Substituting the given values into Eq. (4), you find
a_{A}=\frac{(0.6 \mathrm{ft} / \mathrm{s})^{2}-0}{2(-1.5 \mathrm{ft})}=-0.12 \mathrm{ft} / \mathrm{s}^{2} \quad \mathbf{a}_{A}=0.120 \mathrm{ft} / \mathrm{s}^{2} \uparrowSubstituting this value into a_C= -4a_{A}, you obtain
\mathbf{a}_{C}=0.480 \mathrm{ft} / \mathrm{s}^{2} \downarrowb. Velocity and change in position of B after 2.5 s. Substituting a_{A} in a_{B} = 2a_{A} gives
a_{B}=2\left(-0.2 \mathrm{ft} / \mathrm{s}^{2}\right)=-0.24 \mathrm{ft} / \mathrm{s}^{2}You can use the equations of constant acceleration to find
REFLECT and THINK: One of the keys to solving this problem is recognizing that since there are two cables, you need to write two constraint equations. The directions of the answers also make sense. If block C is accelerating downward, you would expect A and B to accelerate upward.
