Problem#1
A small particle of mass m is pulled to the top of a frictionless half-cylinder (of radius R) by a cord that passes over the top of the cylinder, as illustrated in Figure 1. (a) If the particle moves at a constant speed, show that F = mg cos θ. (Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times.) (b) By directly integrating , find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder.
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| Fig.1 |
Answer:
(a) The radius to the object makes angle θ with the horizontal, so its weight makes angle θ with the negative side of the x-axis, when we take the x–axis in the direction of motion tangent to the cylinder.
∑Fx = max
F – mg cosθ = 0
F = mg cosθ
(b) We use radian measure to express the next bit of displacement as
dr = Rdθ
in terms of the next bit of angle moved through:
W = ∫F.dr
W = ∫0π/2 (mg cosθ)Rdθ
W = mgR
Problem#2
A light spring with spring constant 1200 N/m is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant 1800 N/m. An object of mass 1.50 kg is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.
Answer:
The same force makes both light springs stretch.
(a) The hanging mass moves down by
x = x1 + x2 = mg/k1 + mg/k2
x = mg(1/k1 + 1/k2)
= (1.50 kg)(9.80 m/s2)(1 m/1200 N + 1 m/1800 N)
x = 2.04 cm
(b) We define the effective spring constant as
k = F/x
k = mg/[mg(1/k1 + 1/k2)]
k = (1/k1 + 1/k2)-1
= (1 m/1200 N + 1 m/1800 N)-1 = 720 N/m
Problem#3
A light spring with spring constant k1 is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant k2. An object of mass m is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.
Answer:
See the solution to problem 2
(a) the total extension distance of the pair of springs is
x = mg(1/k1 + 1/k2)
(b) the effective spring constant of the pair of springs as a system is
k = (1/k1 + 1/k2)-1 = k1k2/(k1 + k2)
Problem#4
Express the units of the force constant of a spring in SI base units.
Answer:
k = F/x = N/m = (kg.m/s2)/m = kg/s2
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