An apparatus of the figure below is designed to study insects at an acceleration of magnitude 940 m/s2 (= 96g). The apparatus consists of a 2.0-m rod with insect containers at either end. The rod rotates about an axis perpendicular to the rod and at its center.

(a) How fast does an insect move when it experiences a radial acceleration of 940 m/s2?

(b) What is the angular speed of the insect?

The centripetal acceleration at the ends of the rod is

r w^2,
where a is the angular velocity in radian/sec and r = 1 meter.
Set that equal to 940 m/s^2 and solve for w. (That will be the answer to b)

For the speed of the insect , v = r w

30.11

To solve this problem, we can use the equations relating radial acceleration, tangential speed, and angular speed.

(a) To find the tangential speed of the insect, we can use the equation:

tangential speed = radial acceleration * radius

Given that the radial acceleration is 940 m/s^2 and the radius of the rod is 2.0 m, we can calculate the tangential speed as follows:

tangential speed = 940 m/s^2 * 2.0 m = 1880 m/s

Therefore, the insect moves with a tangential speed of 1880 m/s when it experiences a radial acceleration of 940 m/s^2.

(b) To find the angular speed of the insect, we can use the equation:

angular speed = tangential speed / radius

Using the tangential speed we calculated in part (a) and the radius of the rod (2.0 m), we can calculate the angular speed as follows:

angular speed = 1880 m/s / 2.0 m = 940 rad/s

Therefore, the angular speed of the insect is 940 rad/s.

To solve this problem, we need to use the concepts of radial acceleration and angular speed.

(a) The radial acceleration can be found using the formula:

aradial = r * ω^2

where:
aradial is the radial acceleration,
r is the radius (half the length of the rod, in this case, r = 2.0 m/2 = 1.0 m),
and ω is the angular speed.

We are given that aradial = 940 m/s^2. Plugging in the values, we can solve for ω:

940 m/s^2 = (1.0 m) * ω^2

Rearranging the equation, we have:

ω^2 = 940 m/s^2 / 1.0 m

ω^2 = 940 rad/s^2

Taking the square root of both sides, we get:

ω ≈ √(940 rad/s^2)

Therefore, the angular speed is approximately equal to the square root of 940 rad/s^2.

(b) The angular speed can also be found using the formula:

ω = v / r

where:
v is the linear speed of the insect,
and r is the radius (half the length of the rod, which is 1.0 m in this case).

Since we have already calculated the angular speed, we can rearrange the equation to solve for v:

v = ω * r

Substituting the values, we have:

v = √(940 rad/s^2) * 1.0 m

Calculating the value, we get:

v ≈ √(940 m^2/s^4)

Therefore, the linear speed of the insect is approximately equal to the square root of 940 m^2/s^4.