A 0.700-kg ball is on the end of a rope that is 0.90 m in length. The ball and rope are attached to a pole and the entire apparatus, including the pole, rotates about the pole's symmetry axis. The rope makes an angle of 70.0° with respect to the vertical as shown. What is the tangential speed of the ball?

I was thinking that it would be .90tan(70)=2.47 m/s^2 but I am wrong so I don't know how to do this...

Let the rope tension be T.

T sin70 = M V^2/R
T cos70 = M g
Now, divide the first equation by the second one.
tan70 = V^2/(R*g)

V^2 = (0.90)(9.8)(2.747)= 24.23 m^2/s^2
V = 4.92 m/s

Your answer does not have the dimensions of velocity, and must depend upon g.

What is the tangential speed of the ball?

thnks but i tried both 24.23m^2/s^2 and 4.92 n neither are right i don't understand whats wrong

Radius should be 0.90Sin(70) that would account for the length of the path that the stone is traveling in. Use the equation above and substitute .90Sin(70) for the radius.

To find the tangential speed of the ball, we can use the formula:

v = ω * r

where v is the tangential speed, ω is the angular velocity, and r is the radius of rotation.

First, let's find the angular velocity:

ω = Δθ / Δt

where Δθ is the change in angle and Δt is the change in time.

In this case, the rope makes an angle of 70.0° with respect to the vertical, so the change in angle is 70.0°.

To find the change in time, we need to know the period of rotation. The period can be defined as the time taken to complete one full revolution around the pole.

T = 2π / ω

where T is the period and ω is the angular velocity.

Since the problem does not provide the period, let's assume that the ball completes one full revolution per second (T = 1 s). We can adjust this assumption later if needed.

Substituting T = 1 s into the formula, we get:

1 s = 2π / ω

Solving for ω:

ω = 2π rad / s

Now that we have the angular velocity, we can find the tangential speed by substituting it into the first formula:

v = ω * r

Given that the length of the rope (r) is 0.90 m, we can calculate the tangential speed (v):

v = (2π rad / s) * 0.90 m

Calculating this gives us the tangential speed of the ball.