Calculating Repeating Configuration Time
The problem involves a ball rotating internally at a rate '$r$' rotations per second and revolving externally around a stationary point 'O' at a rate '$R$' revolutions per second. We are given that '$R < r$' and both motions occur in the same sense. We need to find the time after which a specific configuration repeats.
Understanding Relative Motion
The configuration described is when a point on the ball lies on the line connecting the ball's center to the stationary point O. This specific alignment depends on the rotation of the ball relative to its revolution around O.
Since the ball rotates at rate '$r$' and revolves at rate '$R$' in the same sense, and '$r > R$', the rotation effectively 'overtakes' the revolution. The relative angular speed of the ball's rotation with respect to its revolution around O is the difference between their rates.
Relative speed = (Rotational speed) - (Revolution speed)
Relative speed = \( r - R \) rotations per second.
Deriving the Time Period
The configuration repeats when the point on the ball completes one full cycle relative to the line segment connecting the ball's center to point O. This corresponds to one full relative rotation.
The time period '$T$' is the time taken for one relative rotation. It is the reciprocal of the relative speed.
Time \( T = \frac{1}{\text{Relative speed}} \)
Substituting the relative speed:
Time \( T = \frac{1}{r-R} \) seconds.
Important Questions from Speed Time & Distance (Notes)
A and B have to travel from place P to place Q following the same route in their respective cars. A drives at $60$ kmph while B drives at $80$ kmph. Find the time taken by B to reach place Q if A takes $12$ hrs.
On a straight road, a bus is $60$ km ahead of a car running in the same direction. After $3$ hours, the car is $90$ km ahead of the bus. If the speed of the bus is $45$ km/h, then what is the speed of the car (in km/h)?
A train running at the speed of $90$ kmph crosses a $250$ m long platform in $26$ seconds. What is the length of the train (in m)?
A car covers 4 successive stretches of 3 km each at speed of 10 kmph, 20 kmph, 30 kmph and 60 kmph respectively. The average speed of the car for the entire journey is:
- A car travels a total distance L. It travels half the distance with speed $v_1$ and the other half with speed $v_2$. The average speed of the car is :
