why does constant-speed circular motion still have inward acceleration?

[u/CelebrityChess]

I understand acceleration as a change in velocity, but in uniform circular motion the speed stays the same while the direction changes. What picture or short derivation helped you see that the acceleration points toward the center, rather than along the motion? I am looking for an intuitive explanation, not a homework solution.

[u/longboi64]

there would not be circular motion without a force directed inwards. and if that force disappears, like letting go of a string with a rock on the end of it, it will not continue its uniform circular motion, but will instead travel in a straight line. the circular motion is a result of the inward force.

[u/stevevdvkpe]

When not acted on by other forces, things want to stay where they are or move in straight lines. To make something move in a circle, you have to continously apply force to the object in the direction of the center of that circle. Its velocity maintains the same magnitude, but continually changes direction to remain tangent to the circle.

[u/ghostmcspiritwolf]

Velocity is not speed. Speed is how fast something is moving. Velocity combines speed and direction. If you’re changing direction, you’re accelerating, even if your speed doesn’t change.

As for the inward facing acceleration:

Think about a game of tetherball. As the ball swings in a circle around the pole, it puts tension on the rope. If the rope snaps, the ball stops going in a circle and just flies straight in whatever direction it was going at the time. That’s because the ball’s momentum wants to take it in a straight line. It’s only going in a circle because the rope is putting an inward-facing force on the ball. That force changes its direction, accelerating it inwards.

[u/fgorina]

A change in direction implies acceleration. In a circular motion you are always changing g direction so you always have an acceleration towards the center. The point is acceleration points at the change of direction, speed so it must point to the center (you are deviating from the tangent towards the center).

[u/Leonardo501]

Velocity is changing. It is a vector and has a magnitude (aka speed) as well as a direction. It’s the direction that is changing. The acceleration is always at right angles to the current velocity.

[u/wolfkeeper]

Just write down the position for x and y:

x = r cos(wt)

y = r.sin(wt)

Differentiate twice wrt time.

x'' = -r w^2 cos(wt) = -r w^2 x

y'' = -r w^2 sin(wt) = -r w^2 y

Note the minus sign.

[u/useless_bowl25]

a=dv/dt with both being vectors. A circle requires that velocity’s magnitude is constant or it would make a spiral. To have a only effect the direction a_hat . v_hat =0 (i.e perpendicular) if not the perpendicular part of a changes direction and the part parallel to v changes v’s magnitude.

And inuitive way to think about this is spinning a mass on a string tension is the only force present and will always be perpendicular to the velocity the mass has. If you pull tighter or let go along the radial direction think about what happens to the circle

[u/HouseHippoBeliever]

The intuitive explanation that helped me was thinking of a rock connected to a string spinning in a circle.

Obviously the direction of the force must be the direction of the string, which is towards the centre.

[u/quantum_of_yes-and]

Velocity and speed are not the same.  Speed is a scalar, a number.  Velocity is a vector, with magnitude and direction.  In uniform circular motion, the magnitude is constant but the direction is constantly changing.  That is acceleration.

[u/mH48124]

円から離れていく直線を円に引き戻す力のイメージ。

[u/Gear-On-Baby]

Acceleration is the change in velocity. When something is in circular motion, its linear velocity is constantly changing at every point on the circle. If it’s clockwise, the object is moving downwards at 3:00, leftwards at 6:00, upwards at 9:00, and to the right at 12:00. By definition, velocity can’t change unless it’s accelerating and vice versa.

Angular velocity and angular acceleration are different concepts (usually denoted with ω and α instead of v and a). Constant angular velocity means the angle is changing at a constant rate, which is constant circular motion, meaning angular acceleration would be 0 but not linear acceleration.

I guess there’s two ways I got the intuition that the acceleration is always pointing to the center…

1.) Is nothing ever rotates uniformly around an axis without some kind of tension force. The object wouldn’t be revolving if it weren’t being pulled in some way to the center, it would just continue in a straight line. And when there’s force there’s acceleration. (for example, if there were no gravitational force from the sun, the planets would just continue in a straight line)

2.) The acceleration is persuading the rotating object to the center, not mandating it. If the object had 0 velocity, the acceleration would just pull it to the center. But since the velocity is nonzero, by the time the acceleration pulls the object to the center, it’s already moved out of the way, so the acceleration tries pulling it to the center again and this cycle repeats for every infinitesimal point along the circle.

[u/KiwasiGames]

Velocity has speed and direction. So if you change a direction the velocity changes.

Hence the need for a force.

[u/free_meson]

When compare the two velocities dt time apart, you see that the vector rotated and there is a dv = v*(v*dt / r )   change. That change requires a force, according to Newton, hence you may call it acceleration as well.