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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:

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

and final momentum

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

We easily conclude that the momentum of the marble after collision must be

so that