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Chapter 3: Problem 12

A figure skater glides along a circular path of radius \(5.00 \mathrm{~m}\). If she coasts around one half of the circle, find (a) the magnitude of the displacement vector and (b) what distance she skated. (c) What is the magnitude of the displacement if she skates all the way around the circle?

Short Answer

Expert verified
The displacement after skating halfway around the circle is 10.00 m, the distance skated halfway around the circle is 15.71 m, and the displacement after skating the entire circle is 0 m.

Step by step solution

01

Determine the displacement traveled for half the circle

To determine this, we could use the geometry of a circle. The displacement is equivalent to the diameter of the circle when half the circle is completed. As the radius \( r = 5.00 \) m, then the diameter (and thus displacement) \( d = 2r = 2(5.00) = 10.00 \) m.
02

Determine the distance skated for half the circle

The distance covered when skating along half the circle is half the perimeter (circumference) of the circle. The circumference of a circle is given by \( C = 2\pi r \). Thus the distance she skated \( d = \frac{1}{2} C = \frac{1}{2} 2\pi r = \pi r = \pi(5.00) = 15.71 \) m.
03

Determine the displacement traveled for the full circle

When covering the entire circle, the skater returns to the starting position. The displacement, being the shortest distance between two points, will be zero as the start and end points are the same. Thus, the displacement \( d = 0 \) m.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Displacement in Circular Motion
When we talk about movement along a circular path, such as a figure skater gliding around a rink, 'displacement' refers to the shortest distance between the start and end points, which is a straight line. Imagine the skater starts at one point on the edge of a circle and moves halfway around. The displacement in this case isn't the length of the path taken but the straight-line distance between the start and end points鈥攅ssentially the diameter of the circle.

In our example, with a radius of 5 meters, the diameter (and hence the displacement for half a circle) would be twice the radius: \(d = 2r = 2 \times 5.00 = 10.00\ m\). Now, if the skater completes the full circle and returns to the start, the displacement is zero, since her starting and finishing points are the same. Key takeaway: Displacement focuses on where you start and finish, not how much ground you cover.
Circumference of a Circle
The circumference of a circle is the complete distance around its edge. For a skater traveling along the edge of an ice rink, this is the path they would cover if they went all the way around once. To find the circumference, we use the formula \(C = 2\pi r\), where \(r\) is the radius鈥攖he distance from the center of the circle to any point on its edge.

In our skater's case, with a radius of 5 meters, the entire distance around the circle is \(C = 2\pi \times 5.00 = 31.42\ m\). For half the circle, the distance skated is half of this circumference, which is \(\frac{1}{2} C = \frac{1}{2} \times 31.42 = 15.71\ m\). Remember, circumference measures the length around the circle, whereas displacement is concerned with the direct distance between two points.
Radius and Diameter
The concepts of radius and diameter are fundamental in understanding circular motion. The radius, noted as \(r\), is a line segment from the center of the circle to any point on its boundary. It's essential because it defines the size of the circle, and many other calculations depend on knowing the radius.

The diameter, denoted as \(d\), is a line segment that passes through the center and connects two points on the boundary of the circle. It is exactly double the length of the radius: \(d = 2r\). Returning to our example of the figure skater, if the radius of the circular path is 5.00 meters, the diameter becomes 10.00 meters. Understanding radius and diameter is crucial for calculating both the perimeter (circumference) of a circle and the displacement for a given circular motion, which makes them critical values in the physics of circular paths.

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Most popular questions from this chapter

A rocket is launched at an angle of \(53.0^{\circ}\) above the horizontal with an initial speed of \(100 \mathrm{~m} / \mathrm{s}\). The rocket moves for \(3.00 \mathrm{~s}\) along its initial line of motion with an acceleration of \(30.0 \mathrm{~m} / \mathrm{s}^{2}\). At this time, its engines fail and the rocket proceeds to move as a projectile. Find (a) the maximum altitude reached by the rocket, (b) its total time of flight, and (c) its horizontal range.

A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal range \(R\). (a) At what angle \(\theta\) is the rock thrown? (b) In terms of the original range \(R\), what is the range \(R_{\max }\) the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain.

A bomber is flying horizontally over level terrain at a speed of \(275 \mathrm{~m} / \mathrm{s}\) relative to the ground and at an altitude of \(3.00 \mathrm{~km}\). (a) The bombardier releases one bomb. How far does the bomb travel horizontally between its release and its impact on the ground? Ignore the effects of air resistance. (b) Firing from the people on the ground suddenly incapacitates the bombardier before he can call, "Bombs away!" Consequently, the pilot maintains the plane's original course, altitude, and speed through a storm of flak. Where is the plane relative to the bomb's point of impact when the bomb hits the ground? (c) The plane has a telescopic bombsight set so that the bomb hits the target seen in the sight at the moment of release. At what angle from the vertical was the bombsight set?

A daredevil is shot out of a cannon at \(45.0^{\circ}\) to the horizontal with an initial speed of \(25.0 \mathrm{~m} / \mathrm{s}\). A net is positioned a horizontal distance of \(50.0 \mathrm{~m}\) from the cannon. At what height above the cannon should the net be placed in order to catch the daredevil?

S In a local diner, a customer slides an empty coffee cup down the counter for a refill. The cup slides off the counter and strikes the floor at distance \(d\) from the base of the counter. If the height of the counter is \(h\), (a) find an expression for the time \(t\) it takes the cup to fall to the floor in terms of the variables \(h\) and \(g\). (b) With what speed does the mug leave the counter? Answer in terms of the variables \(d, g\), and \(h .\) (c) In the same terms, what is the speed of the cup immediately before it hits the floor? (d) In terms of \(h\) and \(d\), what is the direction of the cup's velocity immediately before it hits the floor?
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