A 0.200 m long string that has a ball attached to it is spinning around on a flat horizontal plane. It spins over a 116° angle in 0.360 s. What is the centripetal acceleration of the ball if it is in uniform circular motion?
To find the centripetal acceleration of the ball, we'll need to use the formula for centripetal acceleration:
a = (v^2) / r
where:
a is the centripetal acceleration,
v is the velocity of the ball, and
r is the radius of the circular path.
First, let's find the velocity of the ball. We can use the equation for angular velocity:
ω = θ / t
where:
ω is the angular velocity,
θ is the angle covered, and
t is the time taken.
In this case, the angle covered is 116°, which needs to be converted to radians by multiplying it by (π/180) since there are π radians in 180°. The time taken is given as 0.360 s.
θ = 116° × (π/180) ≈ 2.02 radians
ω = 2.02 radians / 0.360 s ≈ 5.61 rad/s
Next, let's find the velocity, which is given by:
v = ω × r
We are given the length of the string, which acts as the radius:
r = 0.200 m
v = 5.61 rad/s × 0.200 m ≈ 1.12 m/s
Now we can calculate the centripetal acceleration:
a = (v^2) / r
a = (1.12 m/s)^2 / 0.200 m ≈ 6.30 m/s²
Therefore, the centripetal acceleration of the ball, when it is in uniform circular motion, is approximately 6.30 m/s².