Finding the Acceleration at Point Q of a Ball Pendulum Problem | Homework Help

[blayman5]

Homework Statement

A heavy ball swings on a string in a circular
arc of radius 2 m.
The two highest points of the ball’s trajec-
tory are Q and Q′; at these points the string is
±28◦ from the vertical. Point P is the lowest
point of the ball’s trajectory where the string
hangs vertically down.
The acceleration of gravity is 9.8 m/s2.
What is the magnitude of the ball’s acceler-
ation at the point Q? Answer in units of
m/s2.

The Attempt at a Solution

Wouldn't the acceleration at Q be 0 because there is no velocity at that point.

 

[Doc Al]

Wouldn't the acceleration at Q be 0 because there is no velocity at that point.

The centripetal acceleration will be zero, but what about the tangential acceleration?

 

[blayman5]

Ah, there would be tangential acceleration. How would you go about finding it? The initial steps.

 

[Doc Al]

Apply Newton's 2nd law. What forces act on the ball? What are their tangential components?

 

[blayman5]

The forces acting on the ball are the tension and the mg.
TcosO=mg+ma. Since I do not know the tension, how can I solve for tangential acceleration?

 

[Doc Al]

Does the tension have a tangential component?

 

[blayman5]

T=mg

mgcos0=ma
a=gcos0?

 

[blayman5]

I did TcosO-mg=ma
T=mg+ma/cos0

but there is no mass given in the problem

 

[Doc Al]

Do this. Draw a careful diagram of the ball and string showing all forces acting on the ball. (There are only two.) Then find the tangential component of each force. (Tangential, in this case, is perpendicular to the string.)

You won't need the mass to find the tangential acceleration. Just call it "m"--it will drop out when you apply Newton's 2nd law.

 

[blayman5]

oh ok i did the process and i got a=gtan0.
If the string broke at point Q, would it undergo projectile motion with a horizontal initial velocity?

 

[Doc Al]

oh ok i did the process and i got a=gtan0.

That is not correct.

If the string broke at point Q, would it undergo projectile motion with a horizontal initial velocity?

What's the speed of the ball at point Q?

 

[blayman5]

there is the tension and its components and mg as the forces on the ball. The component TsinO would be the tangential force. Tcos0=mg. T=mg/cos0 Tsin0=ma (mg/cos0)(sin0)=ma
Where did I misinterpret the problem?



The velocity at point Q is 0m/s

 

[Doc Al]

there is the tension and its components and mg as the forces on the ball.

Right. String tension and mg are the only forces on the ball.

The component TsinO would be the tangential force.

No, Tsinθ is the horizontal component of the tension.

Tcos0=mg.

No, Tcosθ (the vertical component of tension) does not equal mg. It would if the ball were in equilibrium, but it's not.

T=mg/cos0 Tsin0=ma (mg/cos0)(sin0)=ma
Where did I misinterpret the problem?

Find the tangential components of the force on the ball.

The velocity at point Q is 0m/s

Right. So if the ball were let go at that point (the string was cut), what would its motion look like?

 

[blayman5]

ok i got gsin0 and it was correct.
Thanks