Question

A ball swings in a vertical circle at the end of a rope 1.50 m long. When the ball is 36.1° past the lowest point on its way up, its total acceleration is (−22.5 i hat + 20.2 j hat) m/s2.

(b) Determine the magnitude of its radial acceleration.
m/s2

(c) Determine the speed of the ball.
m/s

Determine the velocity of the ball.

Help? :/

Answer 1

I'd start by drawing a diagram of the acceleration given in the question.
Include the direction of the rope in the diagram and work out the component of the acceleration along the direction of the rope, that is the radial acceleration.

Answer 2

I'm confused what the i hat and j hat refer to. I thought those were for coordinate systems. Not sure how to translate them to a circle

Answer 3

ihat is a unit vector in the x direction, jhat is a unit vector in the y direction

Answer 4

basically the question is giving you the x and y components of the acceleration, you need to resolve them along that radial direction

Answer 5

How does that relate to the radius?

Answer 6

the radius is the direction along which you want the components of ax and ay, the angle is given in the question so you know what direction it is

Answer 7

Normally the magnitude of a vector is given by \[\sqrt{a ^{2}+b ^{2}}\] correct?

Would it be the square root of (-22.5)^2 + 20.2^2

Answer 8

that just gives you the magnitued of the total acceleration, there's no guarantee that the acceleration lies along the radial direction so that would not be correct

Answer 9

look at my diagram again for reference
and just think about the y component of the acceleration
that y component can be resolved along the radial direction in my diagram, do you see what it will be ?

Answer 10

Im not sure what you mean by resolved.

Answer 11

If you have a vector, you can take any direction you like and ask 'what is the component of the vector in this direction ?'
so I'm just asking, if you have a vector along the y direction, what is it's component along the radial direction in my diagram

Answer 12

The radius is given as 1.5m? I'm not really following you sorry. This is the first problem like this I've ever done and I don't really understand. The example my professor used had r hat and theta hat.

Answer 13

that is fine, rhat is just a vector along the radial direction and theta hat is perpendicular to it

let me ask you another question, suppose a vector p makes an angle theta with the x axis, what is the x component of the vector in terms of p and theta ?

Answer 14

p cos theta

Answer 15

right - and that's all we're trying to do here, we just want the component of a along the radial direction , easier to just get the component of ay along the radial direction first

Answer 16

the question gives you ay, right, it's the bit with k hat, so that points vertically up
and you know that the radial direction makes an angle 36.1 degrees with the vertical . . .

Answer 17

so it would be 1.5cos(36.1)

Answer 18

1.5 is just the length of the rope, you need the 'length' of the acceleration vector as given in the question

Answer 19

or rather, the y component, for the moment

Answer 20

Ok, so use the given j hat value of 20.2 to solve for the radial value? 20.2cos36.1

Answer 21

yes i think that is right

Answer 22

then you need to add to that the contribution from the x component of acceleration along the radial direction as well

Answer 23

Which is -22.5sin(36.1)

Answer 24

yes, but you don't need the minus sign, if you look at the diagram it is going to add to the contribution from ay, not oppose it

Answer 25

sorry i know this is confusing

Answer 26

both bits will add to the inward radial acceleration

Answer 27

anyway, add up those two pieces and you've got the radial acceleration

Answer 28

tell me what you get

Answer 29

29.58

Answer 30

that's what i have
now, the ball is moving in a circle, so there's a formula that relates the speed to the radial acceleration, do you know it ?

Answer 31

No. I just know angular speed formula but that requires knowing the time which is not given.

Answer 32

if something is moving in a circle with speed v, then it must have a radial acceleration equal to v squared divided by the radius of the circle

Answer 33

so have to solve for V?

Answer 34

\[a _{r}=\frac{ v^2 }{ r }\]
it's a very important formula for circular motion

Answer 35

directed towards the centre of the circle, by the way

Answer 36

yes, we calculated ar already, so you can solve this equation to find v