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A steel ball of mass 110 grams moves with a speed of 30 cm / second around a circle of radius 20 cm.
What are the magnitude and direction of the centripetal acceleration of the ball?
The acceleration of an object going at a constant speed around a circle is
a = v^2 / r ,
and the acceleration is directed toward the center of the circle
Therefore:
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a = (30 cm/s)^2 / 20cm = 45 cm/s^2
What are the magnitude and direction of the centripetal force required to keep it moving around this circle?
Net force = mass * acceleration, and the centripetal force is the net force.
Thus
centripetal force = mass * centripetal acceleration.
The object's mass is
110g = .110kg, and 45 cm/s^2 = .45 m/s^3 so
F_cent = m * a_Cent =
.110kg * .45 m/s^2 = 49.5 kg * m/ s^2 = 49.5 N
Since net force = mass * acceleration, an acceleration toward the center implies a net force toward the center.
A steel ball of mass 60 grams, moving at 80 cm / sec, collides with a
stationary marble of mass 20 grams. As a result of the collision the steel ball
slows to 50 cm / sec and the marble speeds up to 70 cm / sec.
Is the total momentum of the system after collision the same as the total
momentum before?
The momentum of a mass m moving at velocity v is p = m v.
We therefore have total initial momentum
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( 0.06 kg * .8 m/s) + (.02 kg * 0 m/s) = 0.048 kg*m/s,
and final momentum
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( 0.06 kg * .5 m/s) + (.02 kg * .7 m/s) = (0.03 kg*m/s) + (0.014 kg*m/s)
= 0.044 kg*m/s
The total momentum before is greater than the total momentum after the
collision.
This cannot happen in a isolated system, so we conclude that either our
measurements are subject to some uncertainty, or the two-mass system is not
isolated.
What would the marble velocity have to be in order to exactly conserve
momentum, assuming the information given for the steel ball is accurate?
Assuming that the steel ball still has after-collision momentum .03 kg*m/s
and before-collision momentum .048 kg m/s, we have
-
after-collision total momentum = .03 kg m/s + marble momentum = .048 kg
m/s
We easily conclude that the momentum of the marble after collision must be
-
.048 kg*m/s - .03 kg*m/s = .018 kg*m/s
so that
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.02 kg * v_marble= .018 kg m/s and
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v_marble = .018 kg m/s / (.02 kg) = 0.9 m/s.