A diver can reduce her moment of inertia by a factor of about 3.5 when changing from a straight position to a tuck position. If she makes two rotations in 1.5s when in the tuck position, what is her rotational speed (rot/s) when in the straight position?
Angular momentum, which is proportional to the product of the Moment of Inertia and speed of rotation, remains constant. Her rotation rate will decrease by a factor of 3.5 in the straight position.
The rate will be (2/1.5)(1/3.5) rot/s
.38rad/s
To solve this problem, we can use the principle of conservation of angular momentum.
The moment of inertia (I) is given by the formula:
I = m * r^2
where:
m is the mass of the object (in this case, the diver)
r is the distance between the axis of rotation and the mass
Let's assume that the diver's moment of inertia in the straight position is I1 and in the tuck position is I2.
We are given that the diver reduces her moment of inertia by a factor of about 3.5 when changing from a straight position to a tuck position, so we can write:
I2 = I1 / 3.5
The angular momentum (L) is given by the formula:
L = I * ω
where:
L is the angular momentum
I is the moment of inertia
ω is the rotational speed (in rad/s)
In the tuck position, the diver makes two rotations in 1.5s. This means that the angular displacement (θ) is 2π (since one full rotation is equal to 2π rad) and the time (t) is 1.5s. So we have:
θ = 2π
t = 1.5s
We can use these values to calculate the angular speed (ω2) in the tuck position:
ω2 = θ / t
ω2 = 2π / 1.5
ω2 ≈ 4.1888 rad/s
Now we can use the principle of conservation of angular momentum to find the rotational speed (ω1) in the straight position:
L1 = L2
Since L = I * ω, we have:
I1 * ω1 = I2 * ω2
Substituting the values we have:
I1 * ω1 = (I1 / 3.5) * 4.1888
Simplifying the equation:
ω1 = (I1 / 3.5) * 4.1888 / I1
ω1 = 1.1968 / 3.5
ω1 ≈ 0.3421 rad/s
Therefore, the diver's rotational speed when in the straight position is approximately 0.3421 rad/s.
To determine the diver's rotational speed when in the straight position, we can use the principle of conservation of angular momentum. Angular momentum (L) is equal to the product of moment of inertia (I) and rotational speed (ω) of the diver:
L = I * ω
According to the problem, the diver can reduce her moment of inertia by a factor of about 3.5 when changing from a straight position to a tuck position. This means that I_tuck = (1/3.5) * I_straight.
We are given that when in the tuck position, the diver makes two rotations in 1.5 seconds. This means that the angular displacement (θ) is 2π radians (two complete rotations) and the time taken (t) is 1.5 seconds.
We can use the formula for angular speed:
ω = θ / t
Substituting the values, we have:
ω_tuck = (2π) / 1.5
Now we need to find the rotational speed (ω_straight) when in the straight position. Since angular momentum is conserved, we can equate the angular momentum in the tuck position (L_tuck) to the angular momentum in the straight position (L_straight):
L_tuck = L_straight
(I_tuck * ω_tuck) = (I_straight * ω_straight)
Substituting the relation between I_tuck and I_straight, we have:
((1/3.5) * I_straight * ω_tuck) = (I_straight * ω_straight)
Simplifying, we find:
(1/3.5) * ω_tuck = ω_straight
Now we can calculate the rotational speed when in the straight position using the value of ω_tuck:
ω_straight = (1/3.5) * ω_tuck
Substituting the value of ω_tuck, we can calculate ω_straight by dividing (2π / 1.5) by 3.5.