Centripetal Acceleration Quiz

Test your understanding of why circular motion involves acceleration even at constant speed. These questions cover its direction, magnitude, key relationships, and geometric derivation.

Q1. What is centripetal acceleration in circular motion?

The rate of change of an object's mass

The acceleration associated with the continuous change in velocity's direction

The rate at which an object's speed increases

The force that always points along the path

Centripetal acceleration describes the acceleration associated with changing velocity direction as an object moves along a circular path.

Q2. An object moves around a circle at constant speed. Why does it still have acceleration?

Its velocity direction changes continuously

Its mass changes continuously

Its speed must be increasing

Its radius changes continuously

Velocity is a vector, so changing its direction means velocity changes even when its magnitude, or speed, stays constant.

Q3. In uniform circular motion, which way does centripetal acceleration point?

Along the instantaneous direction of motion

Away from the centre of the circle

Toward the centre of the circle

Parallel to the radius and away from the path

Centripetal acceleration points inward, toward the centre of the circular path.

Q4. What is the direction of centripetal acceleration relative to the instantaneous velocity?

Perpendicular to it

In the same direction

Opposite to it

At an angle of ##45^\circ## to it

The instantaneous velocity is tangent to the circle, while centripetal acceleration points toward the centre, making the two directions perpendicular.

Q5. Which equation gives the magnitude of centripetal acceleration in terms of speed ##v## and radius ##r##?

##a_c = vr##

##a_c = \dfrac{r}{v^2}##

##a_c = \dfrac{v^2}{r}##

##a_c = \dfrac{v}{r^2}##

The magnitude of centripetal acceleration is given by ##a_c = \dfrac{v^2}{r}##.

Q6. If an object's speed doubles while its circular path radius stays the same, how does its centripetal acceleration change?

It doubles

It becomes four times as large

It is halved

It remains unchanged

Since ##a_c = \dfrac{v^2}{r}##, doubling ##v## multiplies the acceleration by ##2^2 = 4## when ##r## is unchanged.

Q7. If the radius doubles while speed stays constant, what happens to centripetal acceleration?

It doubles

It becomes four times as large

It is halved

It remains unchanged

Centripetal acceleration is inversely proportional to radius at fixed speed, so doubling ##r## halves ##a_c##.

Q8. What are the SI units of centripetal acceleration?

##\mathrm{m/s}##

##\mathrm{m/s^2}##

##\mathrm{kg\,m/s}##

##\mathrm{m^2/s}##

Acceleration has SI units of metres per second squared, written as ##\mathrm{m/s^2}##.

Q9. Which geometric idea is used in the classical derivation of centripetal acceleration?

The circle triangle and velocity triangle are similar

The radius and velocity triangles have equal areas

The velocity triangle is always a right triangle with equal sides

The chord length equals the speed

The derivation compares the triangle formed by two radii and a chord with the triangle formed by two velocity vectors and their difference. These triangles are similar.

Q10. For a small angular change ##\Delta\theta## measured in radians, what is the approximate chord length on a circle of radius ##r##?

##\dfrac{\Delta\theta}{r}##

##r\Delta\theta##

##\dfrac{r}{\Delta\theta}##

##r+\Delta\theta##

For a small angle measured in radians, the chord length is approximately ##r\Delta\theta##.

Q11. For a small change in direction ##\Delta\theta## measured in radians, at constant speed ##v##, what is the approximate magnitude of the change in the velocity vector?

##\dfrac{v}{\Delta\theta}##

##v+\Delta\theta##

##v\Delta\theta##

##\dfrac{\Delta\theta}{v}##

At constant speed, the initial and final velocity vectors have equal magnitudes. For a small change in direction measured in radians, the velocity-vector triangle gives ##|\Delta\vec{v}|\approx v\Delta\theta##.

Q12. How does dividing the small velocity change by the time interval help derive centripetal acceleration?

It gives the rate of change of velocity, which is acceleration

It gives the object's mass

It gives the circle's circumference

It removes the dependence on velocity

Acceleration is the rate of change of velocity. Combining ##\Delta v \approx v\Delta\theta## with the angular change over time leads to the centripetal acceleration magnitude.

Q13. Which relation connects linear speed ##v##, radius ##r##, and angular velocity ##\omega## in circular motion?

##v = \dfrac{\omega}{r}##

##v = r\omega##

##v = r+\omega##

##v = \dfrac{r}{\omega}##

The lesson uses ##v = r\omega## to connect linear speed with angular velocity and radius.

Q14. Using ##v = r\omega##, which expression for centripetal acceleration follows from ##a_c = \dfrac{v^2}{r}##?

##a_c = \dfrac{\omega}{r}##

##a_c = r\omega##

##a_c = r\omega^2##

##a_c = \dfrac{r}{\omega^2}##

Substituting ##v = r\omega## into ##a_c = \dfrac{v^2}{r}## gives ##a_c = \dfrac{(r\omega)^2}{r} = r\omega^2##.

Q15. Two objects have the same speed, but one follows a circle with a smaller radius. Which has greater centripetal acceleration?

The object on the smaller-radius path

The object on the larger-radius path

Both have the same acceleration

Neither has acceleration because their speeds are constant

At the same speed, ##a_c = \dfrac{v^2}{r}## is greater when the radius is smaller.

Q16. A car rounds a bend at steady speed. Which statement best describes its motion?

It has no acceleration because its speed is steady

Its velocity changes direction, so it has acceleration

Its velocity is constant because its speed is constant

Its acceleration points along the road ahead

As the car rounds the bend, its velocity direction changes continuously. That directional change means it has centripetal acceleration.

Q17. A stone is whirled on a string in a circular path. What does the direction of centripetal acceleration indicate?

The direction toward the circle's centre

The direction the stone's speed is increasing

The direction away from the string

The direction of the stone's instantaneous velocity

Centripetal acceleration is directed toward the centre of the circular path, regardless of the object's instantaneous direction of travel.

Q18. Which statement about speed and velocity in uniform circular motion is correct?

Both speed and velocity remain constant

Speed changes while velocity direction remains constant

Speed remains constant, but velocity direction changes

Neither speed nor velocity changes

Uniform circular motion can have constant speed, but velocity changes because its direction continually changes.

Q19. What does the velocity triangle represent in the geometric derivation?

The change between two velocity vectors

The distance travelled in one full revolution

The mass multiplied by the radius

The angle between the path and the centre

The velocity triangle is formed from the two velocity vectors and their difference, which represents the change in velocity.

Q20. For an object moving at the same angular velocity, how does centripetal acceleration depend on radius?

It is proportional to radius, as in ##a_c = r\omega^2##

It is inversely proportional to radius

It is independent of radius

It is proportional to the square of radius

Using ##v = r\omega## in the centripetal acceleration formula gives ##a_c = r\omega^2##. At fixed angular velocity, it is proportional to radius.