NDA (II) 2017 General Ability Test · Question 69 of 150

Correct — A, The centripetal acceleration of the object is smaller for a gentle curve than that for a sharp curve. For motion in a circle at constant speed the centripetal acceleration is the square of the speed divided by the radius. Hold the speed fixed, as the question does, and the acceleration varies inversely with the radius. A gentle curve has a large radius, so dividing by a larger number gives a smaller acceleration; a sharp curve has a small radius and therefore a larger one. Everyday experience says the same thing — take a hairpin bend at the speed you would use on a motorway curve and the sideways pull on you is far greater, which is why sharp bends carry lower speed limits and why a tight turn is the one that skids.

• (b)The centripetal acceleration is greater for a gentle curve than that for a sharp curve. — This inverts the relationship. It would require the acceleration to grow with the radius, whereas the radius sits in the denominator, so a bigger radius means a smaller acceleration.
• (c)The centripetal acceleration is the same for both, the gentle and sharp curves. — This would only be true if the radius made no difference, and it does. The speed is fixed by the stem, so the two curves differ in exactly the quantity that decides the acceleration.
• (d)The centripetal acceleration causes the object to slow down. — Centripetal acceleration points toward the centre of the circle, at right angles to the velocity, so it changes only the direction of motion and not its magnitude. A force perpendicular to the motion does no work, which is exactly why the speed can stay constant while the direction turns.

Acceleration is the rate of change of the velocity vector, and a vector can change by turning without changing length. An object going round a circle at constant speed is accelerating all the time, because its direction is changing all the time, and that acceleration is directed toward the centre. It equals the square of the speed divided by the radius, and the force producing it is supplied by whatever is doing the constraining — friction for a car, tension for a stone on a string, gravity for a satellite.

Two ideas decide this item. First, that speed can be constant while velocity is not, which is what makes option (d) wrong. Second, that the radius is in the denominator, which is what makes (a) right and (b) wrong. The same relation explains a great deal of engineering — roads are banked so that a component of the normal reaction helps supply the centripetal force, railway curves are given a cant for the same reason, and the required speed limit falls as the curve gets tighter. There is no outward force in this picture; the sensation of being flung outward is inertia, felt in the turning frame.

• Centripetal acceleration equals the square of the speed divided by the radius of the path.
• At constant speed, a smaller radius means a larger centripetal acceleration.
• The acceleration points toward the centre and is perpendicular to the velocity.
• Because it is perpendicular to the motion it does no work and cannot change the speed.
• The centripetal force is supplied by friction, tension, gravity or a normal reaction, depending on the situation.

• Reading a gentle curve as the more demanding one; it is the sharp curve that needs the larger acceleration.
• Believing an outward centrifugal force acts on the object in the ground frame.
• Assuming any acceleration must change the speed.

NDA asks this in words rather than with numbers, so the mark goes to whoever can read the relation between speed, radius and acceleration off the formula.

• practice — not a real PYQ

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

  • (a)four times as large
  • (b)twice as large
  • (c)half as large
  • (d)unchanged

  Answer(a) four times as large — the acceleration goes as the square of the speed, so doubling the speed multiplies it by four.

• practice — not a real PYQ

  For a car turning on a level road, the centripetal force is supplied by

  • (a)friction between the tyres and the road
  • (b)the weight of the car
  • (c)the engine's driving force
  • (d)air resistance

  Answer(a) friction between the tyres and the road — which is why a wet or icy surface makes a turn dangerous.