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Circular motion

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Introduction to circular motion

The velocity of a particle points along the tangent to the path it is following. This is shown in the Calculus Primer ( not yet published).

Figure 1

In the drawing above, the instantaneous velocity of a particle following a curved path is shown at one instant.

A particle moves in a circle with constant speed. Its velocity is not constant of course, since the direction of the velocity keeps changing. The velocity is shown at 3 instants in the drawing below.

Figure 2

We set up a coordinate system so that the center of the circle is at the origin. The radius is R and the particle’s angular velocity is ω. Angular velocity is the rate at which the angle that the radius   makes with the x-axis is changing.


Here are some questions to test your conceptual understanding of circular motion:

1. Show that ω R = v, where v is the linear speed of the particle.

Answer

Since v is the uniform linear speed of the particle, it takes a time,

t = 2 π R/v

to complete the circle once. Since its angular speed is ω radians/s, the time it takes to go around the circle once can also be written as,

t = 2π/ω

Thus we get,

2 π R/v = 2π/ω

or

v = ω R

2. In terms of the unit vectors i and j, write down the velocity of the particle when it is at (R,0).

Answer

Since the velocity is always tangential to the path, at (R,0), the velocity is given by

v = v j

where v is the uniform speed.

3. Write down its velocity when it is at (0,R). How much time has elapsed between the two instants?

Answer

The velocity at (0,R) is -v i. The particle has covered a quarter of the circle. Since its speed is uniform, the time elapsed is

Δt = 2 π R/(4v) = π R/(2v) = π R/( 2 ω R) = π/( 2 ω)

4. What is the change in the velocity? Would you say that the particle has accelerated, even though its speed is constant?

Answer

The change in velocity is

Δv = -v i - v j

If the net force on a particle is zero, it travels in a straight line at a constant speed. The moment we see that it is not going on a straight line, we know that it is accelerating and a force is acting on it.

5. What was its average acceleration during this period? Remember that acceleration is a vector. Write the answer in terms of its components. What is the direction of the acceleration?

Answer

Its average acceleration is given by

Δv/Δt = -2ωv (i + j)/π

It points in the direction shown by the black arrow below.

Figure 3

This is the direction in which a constant force would have to act for the duration given in order to produce the given change in the velocity.

6. Do the same exercise for the instants when the particle is at ( R,0) and (-R,0).

7. Do the same for the instants when the particle is at (R,0) and (R/√2, R/√2). This is the instant when the radius makes an angle of 450.

You will find that in general different intervals give us different average accelerations. The acceleration is not constant.

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