- #1
Saitama
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Homework Statement
Calculate the acceleration of the block B in the figure, assuming the surfaces and the pulleys ##P_1## and ##P_2## are all smooth and pulleys and string are light.
(The mass of block C is m)
Ans: F/(7m)
Homework Equations
The Attempt at a Solution
I measured distances from a fixed wall on the right of B. The distances are shown in the second attachment.
Writing down the expression for length of string, (##R## is the radius of pulleys)
$$L=x_{P1}-x_B+\pi R+x_{P1}-x_{P2}+\pi R+x_{C}-x_{P2}$$
Differentiating twice with respect to time,
$$0=2\ddot{x_{P1}}-\ddot{x_B}-2\ddot{x_{P2}}+\ddot{x_{C}}$$
##\because \ddot{x_{P1}}=\ddot{x_A}## and ##\ddot{x_{P2}}=\ddot{x_{B}}##
$$2\ddot{x_A}=3\ddot{x_B}-\ddot{x_C} (*)$$
Assume that the tension in the string is T. Applying Newton's second law for A,
$$F-2T=2m\ddot{x_A} (**)$$
For B,
$$3T=4m\ddot{x_B} (***)$$
For C,
$$T=m\ddot{x_C} (****)$$
Solving the four equations,
[tex]\ddot{x_B}=\frac{3F}{13m}[/tex]
Any help is appreciated. Thanks!