The conical pendulum consists of a bob of mass m attached to the end of an inflexible light?

[匿名]

2017-03-19 12:50 pm

string tied to a fixed-point O and swung around so that it describes a circle in a horizontal plane
while revolving, the string generates a conical surface around the vertical axis collinear with h
the height of the cone (see Figure below).
(a) Find an expression of the angular velocity as a function of g, the acceleration of gravity and
h.
(b) If the number of steady revolutions per minute of a conical pendulum is increased from 70 to
80, what would be the difference in the level of the bob (difference in h)?

回答 (1)

天同

2017-03-19 4:51 pm

(a) Using the symbols given in the diagram,
T.sin(theta) = mg
T.cos(theta) = mr.w^2 where w is the angular velocity of the bob.

Dividing, T.sin(theta)/T.cos(theta) = mg/(mrw^2)
i.e. tan(theta) = g/(r.w^2)
But from the diagram, tan(theta) = h/r
hence, h/r = g/(r.w^2)
w^2 = g/h
or w = square-root(g/h)

(b) 70 rev/min = 70 x 2.pi/60 rad/s = 7(pi)/3 rad/s
and 80 rev/min = 80 x 2.pi/60 rad/s = 8(pi)/3 rad/s

From (a) above, h = g/w^2
Therefore, at 70 rev/min, ho = g/[7(pi)/3]^2 m = 9g/[49(pi)^2] m
At 80 rev/min, h = g/[8(pi)/3]^2 m = 9g/[64(pi)^2] m

Difference in height = ho - h = (9g/pi^2).(1/49 - 1/64) m = 0.0428 m
(Take g = 9.81 m/s^2)

Hence, the level of the bob is raised by 0.0428 m (or 4.28 cm)

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