PROBLEM (1) (20 points) m A block of mass m is connected to a block of mass by a massless string that 2 m passes over a massles and frictionless pulley. The block is connected to an 2
g
ideal vertical spring with spring constant k ! which is fixed to the floor. The spring
m
is initially unstretched and the incline is frictionless. The block m is pulled down ! a distance L stretching the spring the same amount. The block m is then released 1 3 from rest. Assume sin " = and cos " = . Express your answers in terms of 2 2
m 2 k
!
some or all of the given quantities and related constants as needed. (a) (6 pts) Draw a free-body diagram for each block at the moment the block m is released from rest. !
" n
! T
! T
m 2
m
! Fs
!
! m! Fg = g 2
! ! F g = mg
(b) (7 pts) Find the speed of each block when they move the distance L and the spring is unstretched again.
K i + U si + U gi = K f + U sf + U gf
Alternative solution: "K + "U g + "U s = 0
3mv 2f $1 1 m 2' "K = K f # K i = & mv 2f + v f )# 0 = %2 2 2 ( 4 ! % m ( "U g = U gf # U gi = ( mgL sin $ # 0) + ' 0 # gL * & 2 )
Take U g = 0 for a block at the lowest position it has. where K i = 0 and select U gi = 0
! !
! !
1 2 1 1m 2 m kL = mv 2f + v f + mgL sin " # gL 2 2 2 2 2 ! m mgL mgL 2 ! "U g = mgL sin # $ gL = $ =0 !m 3mv f 1 2 1 2 m m 2 2 2 = kL " mgL sin # + gL = kL " gL + gL 4 2 2 2 2 2 1 ! "U s = U gf # U gi = 0 # kL2 ! 2 2 3mv f 1 2 2k 2 2kL ---> v f = L = kL ---> v 2f = 2 2 3m 4 2 3m ! 3mv f 1 2 3mv f 1 2 2k " kL = 0 ---> = kL ---> v f = L 3m 4 2 4 2 ! (c) (7 pts) After!moving the distance ! L the block m continues to move up the incline through a distance d before its speed becomes zero. Find the distance d. Assume that the tension ! in the string becomes ! zero after the blocks!move the distance L and the spring is unstretched again.
1 2kL2 mgd m = 2 3m 2
For block m:
K i + U gi = K f + U gf ; K f = 0 and select U gi = 0
d=
2
1 mgd 2kL where v i2 = mv i2 = mgd sin " = 2 2 3m ! !
!
2kL2 3mg
!
! !
Phys 105 Final Examination
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Friday, 12-August 2011