Centripetal Force and Centripetal Acceleration Questions

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1 Centripetal Force and Centripetal Acceleration Questions A 2.10 m rope attaches a tire to an overhanging tree limb. A girl swinging on the tire has a tangential speed of 2.50 m/s. If the magnitude of the centripetal force is 88.0 N, what is the mass of the girl and the tire? A bicyclist is riding at a tangential speed of 13.2 m/s around a circular track. The magnitude of the centripetal force is 377 N and the combined mass of the bicycle and rider is 86.5 kg. What is the track s radius? A 905 kg car travels around a circular track with a circumference of 3.25 km. If the magnitude of the centripetal force is 2140 N, what is the car s tangential speed? Tarzan tries to cross a river by swinging from one bank to another on a vine that is 10.0 m long. His speed at the bottom of the swing is 8.0 m/s. Tarzan does not know that the vine has a breaking force of 1.0 x 10 3 N. What is the largest mass that Tarzan can have and still make it safely across the river? A 2.00 x 10 3 kg car rounds a circular turn of radius 20.0 m. If the road is flat and the coefficient of static friction between the tires and the road is.70, how fast can the car go without skidding?

2 Centripetal Force and Centripetal Acceleration Newton s First Law tells us that just by virtue of the fact that an object is moving in a circle, we know that an unbalanced force must be acting on it. The above, however, does not really answer the why. The existence of an unbalanced force is perhaps further explained by the following: Direction is changing, so velocity is changing, therefore the object is accelerating and therefore an unbalanced force must be acting on it. The above point does offer a deeper explanation but still. Why does a change in direction mean the velocity is changing? Is the change something that is quantifiable? Meaning, is a number changing when direction changes? To answer this question, we turn our attention to the activity on the following screen:

3 An object is experiencing uniform circular motion (moving in a circle at a constant speed). The dots below represent the position of the object a 8 different instants as it travels in a circle. Assuming that the object is travelling in the clock-wise direction, draw a velocity vector representing the direction of the object s velocity at every instant.

4 Looking at the drawing you have in your notes, which hopefully matches the adjacent diagram, why do you think the velocity of the object as it moves around a circle is referred to as the tangential velocity? The direction of the object s velocity at any point on the circle is tangent to the circle at that point

5 So the direction of the object s velocity at any point on the circle is tangent to the circle at that point. But back to the original question: Is a change in direction something that is quantifiable? Meaning, when direction changes is any number that defines velocity changing? To answer this question, draw a diagram just consisting of the velocity vectors representing the direction of the object at every instant as it moves in a circle (as you see below on the right). Then draw in the x and y components for all those vectors. Note: the diagram on the right is identical to that on the left minus the circle.

6 A H B G C F D E You can clearly see in the adjacent diagram that at one point the velocity vector has an x- component equal to 0 (e.g., A and E), and in the next instant there is a nonzero x-component. A number has changed! We can also see in the transition from F to G and B to C, the y- component goes from some nonzero number to 0. Again, a number has changed!

7 The Direction of F c and a c We know if not for the centripetal force, an object would not move in a circle. We can see from this diagram, that if the centripetal force was to disappear at any instant, the object would cease moving in a circle and would fly off in a straight line in a path that was tangent to the circle at the point that the force disappeared. So let s answer the question: What is the direction of the centripetal force? Hint: The force is directed towards the same point at every instant. The centripetal force (and hence a c ) is always directed towards the center of the circle.

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