A 0.125 kg meter stick is rotating (as shown to the right) in a horizontal plane about an axis through its center at 3.45 rad/s. a) Find the rotational inertia of this stick. At one of its ends, find its b) speed and c) centripetal acceleration. If this stick comes to rest in 4.27 s, find d) its angular acceleration and e) the net torque acting on it.
To solve this problem, we can use the formulas related to rotational motion:
a) The formula for rotational inertia of a slender rod rotating about an axis through its center is given by:
I = (1/12) * m * L^2
where m is the mass of the rod and L is its length. In this case, m = 0.125 kg and L is not given. So, we need to know the length of the meter stick to calculate its rotational inertia.
b) The speed of the end of the meter stick can be calculated using the formula:
v = ω * r
where ω is the angular velocity and r is the distance of the end of the meter stick from the axis of rotation. Since we don't have r, we cannot calculate the speed.
c) The centripetal acceleration can be calculated using the formula:
a = ω^2 * r
where a is the centripetal acceleration, ω is the angular velocity, and r is the distance of the end of the meter stick from the axis of rotation. Since we don't have r, we cannot calculate the centripetal acceleration.
d) The angular acceleration can be calculated using the formula:
α = Δω / Δt
where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the change in time. In this case, Δω is the final angular velocity (0 rad/s) minus the initial angular velocity (3.45 rad/s), and Δt is 4.27 s.
e) The net torque acting on the meter stick can be calculated using the formula:
τ = I * α
where τ is the net torque, I is the rotational inertia (which we need to calculate in (a)), and α is the angular acceleration (which we calculated in (d)).
Unfortunately, without knowing the length and distance from the axis of rotation, we cannot calculate b), c), and e).
a) To find the rotational inertia of the meter stick, we need to use the formula for the rotational inertia of a thin rod rotating about its center.
The formula for the rotational inertia of a thin rod rotating about its center is given by:
I = (1/12) * M * L^2
Where:
I = rotational inertia
M = mass of the rod
L = length of the rod
In this case, the mass of the meter stick is given as 0.125 kg. Let's assume the length of the meter stick is L.
So, the rotational inertia of the meter stick is:
I = (1/12) * 0.125 kg * L^2
b) To find the speed of the meter stick at one of its ends, we can use the formula for the linear speed of an object rotating in a circle.
The linear speed of an object rotating in a circle is given by:
v = ω * r
Where:
v = speed
ω = angular velocity
r = radius of rotation
In this case, the angular velocity (ω) is given as 3.45 rad/s, and the radius of rotation (r) is L/2 since the stick rotates about its center.
So, the speed is:
v = 3.45 rad/s * (L/2)
c) To find the centripetal acceleration at one of the ends, we can use the formula for centripetal acceleration.
The centripetal acceleration is given by:
a = ω^2 * r
Where:
a = centripetal acceleration
ω = angular velocity
r = radius of rotation
In this case, the angular velocity (ω) is given as 3.45 rad/s, and the radius of rotation (r) is L/2.
So, the centripetal acceleration is:
a = (3.45 rad/s)^2 * (L/2)
d) To find the angular acceleration, we can use the formula for angular acceleration given by:
α = ωf - ωi / t
Where:
α = angular acceleration
ωi = initial angular velocity
ωf = final angular velocity
t = time
In this case, the initial angular velocity (ωi) is 3.45 rad/s, the final angular velocity (ωf) is 0 (since it comes to rest), and the time (t) is 4.27 s.
So, the angular acceleration is:
α = (0 - 3.45 rad/s) / 4.27 s
e) To find the net torque acting on the meter stick, we can use the formula for torque given by:
τ = I * α
Where:
τ = net torque
I = rotational inertia
α = angular acceleration
In this case, we already know the rotational inertia (I) from part a) and the angular acceleration (α) from part d).
So, the net torque is:
τ = (1/12) * 0.125 kg * L^2 * [(0 - 3.45 rad/s) / 4.27 s]
By substituting the given values, you can calculate the respective answers.