Circular motion and linear speed of an object

In summary, the object is traveling around a circle with a radius of 5cm and sweeping out a central angle of 1/3 radian in 20 seconds. The angular speed of the object is π/60 radians/sec and the linear speed is 1/12 cm/s. The book's answer of 12 m/s is incorrect.
  • #1
ragbash
2
0
An object is traveling around a circle with the radius of 5cm. If in 20sec the central angle of 1/3 radian is swept out, what is the angular speed of the object? Linear speed?

Here's how I did it. angular speed-->
a) omega=theta(in radians)/elapsed time
= π/3/20= π/3*1/20 = π/60 radians/sec. Is that the same as the answer in the book, 1/60?

linear speed-->
b) v=rw (length/radius)(omega=angular speed)
=5cm*(1/60)= 1/12 cm/s
the answer in the book is 12m/s.

what did I do wrong?
 
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  • #2
Where did you get the n from? You went 1/3 radian in 20 seconds. To get the radian speed just find the distance traveled in one second. The book is right.

If it really was 12 m/s it would have gone around the circle about 60 times in 20 seconds since the length of the circumference is 2 * pi * radius or .314 m. So the book is wrong. You are wrong too. Remember a radian is radius / arch distance. So r/s = 1/3 in 20 seconds. Once you have arch distance traveled in 20 seconds, finding linear speed is easy.
 
  • #3
interested learner, I don't think it was "n", it was [itex]\pi[/itex] using an overly simple font.

ragbash, your problem says "1/3 radian". For some reason you used "[itex]\pi/3[/itex] radians.

No, [itex]\pi/3[/itex] is not the same as 1/3!

The circumference of a circle is [itex]2\pi r[/itex] or, since r= 5 here, [itex]10\pi[/itex] cm. Since the object moves 1/60 radian/sec and there are [itex]2\pi[/itex]radians in a circle, it is moving at [itex]\frac{1}{120\pi}[/itex] "circles per second" and so [itex]\frac{1}{120\pi}(10\pi)= 1/12[itex] cm/sec.
Your book apparently has a typo for the second.
 
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Related to Circular motion and linear speed of an object

1. What is circular motion and how is it different from linear motion?

Circular motion is the movement of an object along a circular path. It is different from linear motion, which is movement in a straight line, because the direction of motion in circular motion is constantly changing.

2. How is the speed of an object in circular motion measured?

The speed of an object in circular motion is measured by its linear speed, which is the distance traveled per unit of time along the circular path. It can also be measured by its angular speed, which is the rate of change of its angular position.

3. What factors affect the linear speed of an object in circular motion?

The linear speed of an object in circular motion is affected by the radius of the circular path, the angular speed of the object, and the mass of the object. A larger radius will result in a higher linear speed, while a higher angular speed and a lower mass will also increase the linear speed.

4. How does centripetal force relate to circular motion and linear speed?

Centripetal force is the force that keeps an object moving in a circular path. It is directed towards the center of the circular path and is proportional to the object's mass and the square of its linear speed. In other words, as the linear speed increases, so does the centripetal force.

5. Can an object have a constant linear speed in circular motion?

Yes, an object can have a constant linear speed in circular motion if the magnitude of its centripetal force remains constant. This can be achieved by adjusting the angular speed or radius of the circular path to maintain a balance between the centripetal force and the object's inertia, resulting in a constant linear speed.

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