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The solutions to four mechanics problems involving friction, newton's laws, and potential and kinetic energy. The problems include equations and values for mass, force, acceleration, and distance.
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(a)
F
m (^1)
T
Ffriction
m 1 g
N
m (^2) m 2 g
T
T
T
(b) The block m 1 moves in x-direction:
m 1 a = F cos θ − T − N μ, N = m 1 g − F sin θ
The block m 2 moves in y-direction:
m 2 a = T − m 2 g
Put in the values of m 1 , m 2 , F , and μ, we obtain
3 M a =4M g
(3M g − 12 M g 5
M a =T − M g
Add the above two equation together
4 M a = 2M g
We find a = 12 g. For block m 1 : a points to the right and |a| = a For block m 2 : a points up and |a| = a
(c) T = M a + M g = M g (^32)
(a)
M g
T
T x
y
1
2
(b) Distance between the ball and the rod r =
Speed of the ball v = (^2) Tπr = 6 πLT.
(c) The acceleration a points to the left. Its magnitude is |a| = v 2 r =^
12 π^2 L T 2.
(d) The the x- and y-components of the force on the ball from the upper string: F 1 = (−T 1 35 , T 1 45 ) The the x- and y-components of the force on the ball from the lower string: F 1 = (−T 2 35 , −T 2 45 ) Newtons law for y-direction
0 = T 1
− M g
Newtons law for x-direction −M |a| = −T 1
We find
T 1 − T 2 =M g
T 1 + T 2 =M |a|
or
T 1 = M g
= M g
10 π^2 L T 2
(e)
T 2 = −M g
= −M g
10 π^2 L T 2
(a) c
(b) a
(c) b
(d) e
(e) c
