it is question of velocity and accerlation.

Question

An insect crawls at a constant rate u along the spoke of a cart wheel of radius a,the cart moving with velocity v. Find the acceleration along and perpendicular to the spoke at time t.                      

 

    Its ans. Are -uv²t/a and 2uv/a

Answer 1

Hi Raghav!

I am going to start from a certain point that, given your statement of this problem, it seems you would have come to either in a class or from a textbook (such as Classical Dynamics by Goldstein or Golden Dynamics, by Gupta).  If that is not the case, we can go into how this is derived.  I am also presenting this with an assumption of a knowledge of calculus -- which is usually the case with this classic problem.  Again, if I am mistaken, we can talk about it in algebra terms, with some assumptions, as well.

 

The place I will start is with the statement that, for a particle (such as our point insect) who can move radially and tangentially on a rotating disc, the expression for acceleration is:

 

a = (r'' - r(θ')²)r-hat + (2r'θ'+θ'')θ-hat

 

where I mean the ' to represent what the "dot" over the variable usually represents, differentiation of the quantity with respect to time.

 

 The first two terms in parentheses on the left represent acceleration in the radial direction (along the spoke, in the problem above) and the two terms in parentheses on the right represent acceleration in the tangential direction (perpendicular to the spoke, for us).

 

Note that the two factors that represent acceleration in the radial direction are r'', due to the object changing speed in the radial direction, and -r(θ')², that points radially inward.  Since θ', as I have defined it, represents the angular velocity, this term can be seen to be the centripetal acceleration.

 

The two factors in the tangential direction are θ'', which represents angular acceleration (often denoted α), and 2r'θ'.  The latter term represents the tangential acceleration due to the object moving to a different radius, and thus traveling at a different linear speed on the different-radius circle to stay rotating with the object.  This is directly related to something you may have heard of called the Coriolis effect.

 

In any case, what we have to do is evaluate this for the situation of this problem.  If we reference the insect's motion to the center of the wheel, we can immediate define r' and r'' for the above expression.  Because we can write:

 

r = ut  (just distance = rate*time, with the insect crawling radially along the wheel -- I assume outward; it is only a sign change to make it inward)

 

Thus:

 

r' = u

r'' = 0  (since the insect is moving at a constant radial speed along the spoke and not accelerating in that direction)

 

We also have to determine values for θ' and θ'', and this may be the crux of the problem.  It looks like the problem means to set up a situation where the wheel is rolling without slipping, with translational velocity (along the road) v.  The key connection here is that, if the wheel is rolling without slipping, every little bit the cart moves down the road corresponds to an equal linear distance rolled by a point on the outer circumference of the wheel.  Because of that, we can directly relate the linear velocity of the care down the road and the linear velocity of a point on the outer circumference of the wheel, at radius a.

 

Since v_t = rθ' for circular motion, this means -- in this case:

 

v = aθ' --> θ' = v/a

 

Since v and a are both constants in this problem, it can quickly be seen that θ'' = 0.

 

This just means that if the translational velocity of the wheel is constant, the linear velocity of a point on the rim of the wheel is constant.  Thus, the angular velocity of a point on the rim (and any other point on the wheel, if it is all turning together) is constant, and the angular acceleration is therefore 0.

 

So the above expression for overall acceleration becomes:

 

a = (r'' - r(θ')²)r-hat + (2r'θ'+θ'')θ-hat --> (0 - ut(v/a)²)r-hat + (2u(v/a) + 0)θ-hat

 

Because of that, we can say that the tangential acceleration is the centripetal acceleration, -utv²/a²  (I think that it is supposed to be a² in the denominator here; if there is not, let me know, and I will look at it again).

 

In the radial direction, it is 2uv/a.

 

I hope that helps!  If you have any other questions, just let me know.