Circular Motion Relationships | Show Me The Physics

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How radius and velocity combine to change centripetal acceleration

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Diagram of an object moving in a circle with an incorrectly drawn acceleration vector

What's wrong with this picture?

↓

Acceleration Vector should be directed towards center

Kinematics in 2D

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B. Factors Affecting Centripetal Acceleration

1. Velocity and Acceleration?

Equation:

a_c = V² / r

Direct Square Relationship (Both variables on top, one is squared)

V	a
x 2	x 4
x 3	x 9
x 4	x 16
x 1/2	x 1/4
x 1/4	x 1/16

2. Acceleration and Radius?

Equation:

a_c = V² / r

Man swinging a remote-control car in a circle

Inverse Relationship

(One variable on top, other on bottom)

a	r
x 1/2	x 2
x 1/3	x 3
x 1/4	x 4
x 4	x 1/4
x 2	x 1/2

C. More Word Problems

Given

A 2.0 kg cart travels in a horizontal circle at a constant speed of 6.0 m/s.
If the r of circle is 3.0 m, what is centripetal acceleration?

↓

a_c = V² / r
= (6.0 m/s)² / 3.0 m

a_c = 12 m/s²

Challenge

Given

Calculate the speed an earth satellite must have to maintain a circular orbit at an altitude of 2.0 x 10⁵ m (200,000 m) where the acceleration due to gravity is 9.2 m/s² (radius of earth is 6,400,000 m)

[ Flash ]

↓

r = 6,400,000 m + 200,000 m

r = 6,600,000 m

↓

a_c = V² / r

9.2 m/s² = V² / (6,600,000 m)

9.2 m/s²(6,600,000 m) = V²

V = 7,792 m/s

or 7800 m/s (correct significant digits)

Non-uniform Circular Motion

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Check Your Understanding

5 quick questions — pick an answer to see if you’re right.

1. An object moves in a circle of fixed radius. If its speed doubles, its centripetal acceleration becomes

2. An object keeps the same speed but the radius of its circle is doubled. Its centripetal acceleration becomes

3. A 2.0 kg cart travels in a horizontal circle of radius 3.0 m at 6.0 m/s. What is its centripetal acceleration?

4. A satellite orbits 200,000 m above Earth's surface. Earth's radius is 6,400,000 m. What radius goes into ac = v²/r?

5. If both the speed and the radius are cut in half, the centripetal acceleration becomes

NYSSLS Regents Practice · Circular Motion · Mathematical Relationships

The Astronaut Training Centrifuge

Astronauts train in a centrifuge: a capsule at the end of an arm spins in a horizontal circle of radius 8.0 m. The goal is to produce a centripetal acceleration of 3g (about 29.4 m/s²).

What speed must the capsule have?

▶ Want a challenge? Open 6 more Regents-style questions (harder · mixed formats)

Challenge score: 0 / 6

Multiple choice

C1. Which graph best represents centripetal acceleration versus radius at a constant speed?

Support or Refute · Justify

C2. An engineer makes the following claim.

Claim: To produce 6g (58.8 m/s²) at the same 8.0-m radius, the capsule’s speed must be doubled.

Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.

Support Refute

Substitute values

C3. At 3g the capsule moves at 15.3 m/s. Substitute numerical values into a mathematical representation that can be used to determine the time for one revolution of the centrifuge. Then calculate the period.

T = 2π( m)( m/s) = s

Determine the value

C4. Determine the magnitude of the net force on a 75-kg astronaut in the capsule during the 3g run.

F_net = N

Multiple choice

C5. The capsule’s speed is doubled at the same radius. By what factor does the centripetal acceleration change?

Constructed response

C6. Engineers propose replacing the 8.0-m arm with a 16-m arm. Identify one benefit and one drawback of the longer arm for producing 3g.

Centripetal Force →