When a compact disk with a 12.0-cm diameter is rotating at 4.2 rad/s, what is (a) the linear speed and (b) the centripetal acceleration of a point on its outer rim?
Why did the CD go to therapy? Because it had trouble finding its center! Now, let's solve your problem.
(a) To find the linear speed of a point on the outer rim of the CD, we need to multiply the radius (half of the diameter) by the angular velocity. The radius of the CD is 12.0 cm / 2 = 6.0 cm = 0.06 m. So, the linear speed is given by:
Linear speed = Radius × Angular velocity
Linear speed = 0.06 m × 4.2 rad/s
So, the linear speed is approximately 0.252 m/s.
(b) Now, to find the centripetal acceleration of a point on the outer rim, we can use the formula:
Centripetal acceleration = (Linear speed)² / Radius
First, let's convert the radius to meters:
Radius = 0.06 m
Now, let's solve for the centripetal acceleration:
Centripetal acceleration = (0.252 m/s)² / 0.06 m
Centripetal acceleration is approximately 1.05 m/s².
Keep in mind that these calculations assume the CD is rotating in a circular path without any other forces acting upon it.
To find the linear speed, we can use the formula:
Linear speed (v) = Angular speed (ω) × Radius (r)
a) The angular speed is given as 4.2 rad/s, and the radius is half of the diameter, which is 12.0 cm ÷ 2 = 6.0 cm = 0.06 m.
Linear speed (v) = 4.2 rad/s × 0.06 m
v ≈ 0.252 m/s
Therefore, the linear speed of a point on the outer rim is approximately 0.252 m/s.
b) To find the centripetal acceleration, we can use the formula:
Centripetal acceleration (aᵤ) = Linear speed (v) × Angular speed (ω)
Given that the linear speed is 0.252 m/s and the angular speed is 4.2 rad/s:
Centripetal acceleration (aᵤ) = 0.252 m/s × 4.2 rad/s
aᵤ ≈ 1.058 m/s²
Therefore, the centripetal acceleration of a point on the outer rim is approximately 1.058 m/s².
To find the (a) linear speed and (b) centripetal acceleration of a point on the outer rim of a rotating compact disk, you can use the following formulas:
(a) Linear Speed:
Linear speed refers to the rate at which an object moves along a circular path. It can be calculated using the formula:
Linear Speed = Radius x Angular Speed
In this case, since the diameter of the compact disk is given, we first need to convert it to radius by dividing it by 2:
Radius = Diameter / 2 = 12.0 cm / 2 = 6.0 cm = 0.06 m
Next, the angular speed is given as 4.2 rad/s. Now we can plug these values into the formula to find the linear speed:
Linear Speed = 0.06 m x 4.2 rad/s = 0.252 m/s
Therefore, the linear speed of a point on the outer rim of the compact disk is 0.252 m/s.
(b) Centripetal Acceleration:
Centripetal acceleration is the acceleration experienced by an object moving along a circular path towards the center of the circle. It can be calculated using the formula:
Centripetal Acceleration = (Angular Speed)^2 x Radius
First, let's calculate the square of the angular speed:
Angular Speed^2 = (4.2 rad/s)^2 = 17.64 rad^2/s^2
Now, we can find the centripetal acceleration by plugging this value and the radius (0.06 m) into the formula:
Centripetal Acceleration = 17.64 rad^2/s^2 x 0.06 m = 1.0584 m/s^2
Therefore, the centripetal acceleration of a point on the outer rim of the compact disk is 1.0584 m/s^2.
tangential linear speed= angular speed*radius=4.2 rad/s*12cm= ...
centripetal acceleration= v^2/radius= w^2*radius= 4.2^2 * 12cm/s^2