CDS (I) 2021 General Knowledge · Question 118 of 120

Correct — D, direction perpendicular to velocity. In uniform circular motion the speed never changes but the direction of travel does, continuously, and a change in the direction of velocity is an acceleration just as surely as a change in its size. Because the speed is constant, none of that acceleration can lie along the direction of motion — any component along the velocity would speed the body up or slow it down. What is left is a component at right angles to the velocity, and since the velocity is tangential to the circle, perpendicular to it means along the radius, pointing inwards to the centre. That is the centripetal acceleration, of size v²/r, and the force producing it is whatever physically pulls the body inwards: the string for a whirled stone, gravity for a satellite, friction for a car on a bend.

• (a)direction of motion. — An acceleration along the direction of motion changes the speed. The motion here is uniform, so the speed is fixed and no such component can exist.
• (b)tangent to the circle at the point of observation. — The tangent is the direction of the velocity, not of the acceleration. This option describes where the body is heading, which is the answer to a different question.
• (c)direction of velocity. — Another way of writing option (a), since the velocity points along the direction of motion. It fails for the same reason — it would make the body speed up.

Velocity is a vector, so it can change either in size or in direction, and either change counts as acceleration. Uniform circular motion is the case where only the direction changes. The resulting acceleration points towards the centre of the circle at every instant, has magnitude v²/r, and needs a real inward force to supply it. Remove that force — cut the string — and the body flies off along the tangent, not outwards along the radius.

Three of the four options say the same thing in different words: 'direction of motion', 'tangent to the circle' and 'direction of velocity' all name the tangential direction. Spotting that they are one answer written three ways settles the item, because a well-formed question cannot have three correct options. The reasoning behind it is the part worth keeping: constant speed forbids any tangential component of acceleration, so what remains must be perpendicular. The everyday check is a stone whirled on a string — the hand pulls inwards the whole time, and when the string breaks the stone leaves along the tangent.

• In uniform circular motion the speed is constant while the direction of the velocity changes continuously.
• The acceleration is centripetal — directed towards the centre and perpendicular to the velocity at every instant.
• Its magnitude is v²/r, so a sharper bend at the same speed demands a larger acceleration.
• The centripetal force is supplied by tension, gravity, friction or the normal force, depending on the situation.
• If the inward force is removed the body continues along the tangent, by the first law of motion.

• Assuming that constant speed means zero acceleration; the direction is changing, so it does not.
• Reading the tangential direction as the acceleration, when it is the velocity.
• Expecting a body released from circular motion to fly outwards along the radius rather than along the tangent.

As a direction-of-acceleration item, or as a numerical on v²/r with the radius or the speed changed.

• practice — not a real PYQ

  A stone tied to a string is whirled in a horizontal circle at constant speed. If the string breaks, the stone flies off

  • (a)along the radius, outwards
  • (b)along the tangent at that point
  • (c)along the radius, inwards
  • (d)vertically upwards

  Answer(b) along the tangent at that point — with the inward force gone, the stone keeps the velocity it already had, and that velocity is tangential.

• practice — not a real PYQ

  If the speed of a body in uniform circular motion is doubled while the radius is unchanged, its centripetal acceleration becomes

  • (a)half
  • (b)double
  • (c)four times
  • (d)unchanged

  Answer(c) four times — the acceleration is v²/r, so it varies as the square of the speed.