system consists of a steal beam of mass M 􏰀 68.4 kg and length L 􏰀 1.84 m, with a 39.0-kg

brass sphere attached at the right end of the rod, as shown below.
BA
a. Determine the moment of inertia of the system for rotation about an axis through the center of the steel beam (axis A).
b. Determine the moment of inertia of the system for rotation about an axis through one end of the beam (axis B).
c. Determine the constant horizontal force, F, that must be exerted on the brass sphere in order to produce an angular acceleration of the system such that 􏰅 􏰀 2􏱐 rads. F is exerted perpendic- ular to the steel beam, and the system rotates around axis B.
d. Consider part c. If the system started from rest, determine the linear velocity of the brass sphere after the force has been applied for 4.0 s.

125, 130

I know this question is old. But for anyone looking it up now, here's how to do it.

Use the formula
I = 1/12ml^2

a. To determine the moment of inertia of the system for rotation about an axis through the center of the steel beam (axis A), we need to find the moment of inertia of the steel beam and the moment of inertia of the brass sphere.

The moment of inertia of the steel beam can be calculated using the formula for a rod rotating about its center:
I_beam = (1/12) * M_beam * L^2

Where:
M_beam = mass of the steel beam = 68.4 kg
L = length of the steel beam = 1.84 m

Substituting the given values into the formula, we get:
I_beam = (1/12) * 68.4 kg * (1.84 m)^2

b. To determine the moment of inertia of the system for rotation about an axis through one end of the beam (axis B), we need to consider the moment of inertia of the steel beam, the moment of inertia of the brass sphere, and the parallel axis theorem.

The moment of inertia of the steel beam about axis B is the same as the moment of inertia about axis A, which we calculated in part a.

The moment of inertia of the brass sphere about its center can be calculated using the formula for a solid sphere rotating about its center:
I_sphere = (2/5) * M_sphere * R^2

Where:
M_sphere = mass of the brass sphere = 39.0 kg
R = radius of the brass sphere (assuming it's a perfect sphere, R = 1/2 of the length of the steel beam)

Substituting the given values into the formula, we get:
I_sphere = (2/5) * 39.0 kg * ((1.84 m)/2)^2

Using the parallel axis theorem, we can calculate the moment of inertia of the brass sphere about axis B:
I_Brass_sphere = I_sphere + M_sphere * d^2

Where:
d = distance between axis A and axis B (equal to half the length of the steel beam)

Substituting the values into the formula, we get:
I_Brass_sphere = I_sphere + 39.0 kg * ((1.84 m)/2)^2

c. To determine the constant horizontal force, F, required to produce an angular acceleration of the system, we can use the equation:
τ = I_total * α

Where:
τ = torque produced by the force F,
I_total = total moment of inertia of the system,
α = angular acceleration of the system.

Since the force F is exerted perpendicular to the steel beam, the torque τ can be calculated as:
τ = F * d

Where:
d = distance between axis A and axis B (equal to half the length of the steel beam)

Setting these equations equal to each other, we get:
F * d = I_total * α

Solving for F:
F = (I_total * α) / d

Substituting the values of I_total calculated in part a and part b:
F = (I_beam * α) / d

Substituting the given values into the formula, we get:
F = (I_beam * 2 rad/s^2) / ((1.84 m) / 2)

d. To determine the linear velocity of the brass sphere after the force has been applied for 4.0 seconds, we can use the equation:
v = ω * R

Where:
v = linear velocity of the brass sphere,
ω = angular velocity of the system,
R = radius of the brass sphere (assuming it's a perfect sphere, R = 1/2 of the length of the steel beam).

The angular velocity ω can be calculated using the formula:
ω = α * t

Where:
α = angular acceleration of the system = 2 rad/s^2
t = time = 4.0 seconds

Substituting the values into the formula, we get:
ω = 2 rad/s^2 * 4.0 s

Finally, substituting the calculated value of ω and the given value of R into the formula for linear velocity, we can find the answer for part d.

In order to solve this problem, we need to use the equations for moment of inertia, torque, and angular acceleration. Here's how we can approach each part of the problem:

Part a:
The moment of inertia (I) of the system about an axis through the center of the steel beam (axis A) can be calculated by adding the individual moments of inertia of the steel beam (I_beam) and the brass sphere (I_sphere).

The moment of inertia of a slender rod about its center is given by the formula: I_beam = (1/12) * m_beam * L_beam^2, where m_beam is the mass of the steel beam and L_beam is its length.

The moment of inertia of a sphere about its center is given by the formula: I_sphere = (2/5) * m_sphere * R^2, where m_sphere is the mass of the brass sphere and R is its radius.

So, you can calculate the moment of inertia of the system about axis A by plugging in the given values for m_beam and L_beam, and applying the formula for both the beam and the sphere.

Part b:
The moment of inertia of the system about an axis through one end of the beam (axis B) can be calculated by adding the individual moments of inertia of the steel beam (I_beam) and the brass sphere (I_sphere) but with respect to axis B.

The moment of inertia of a slender rod about one end is given by the formula: I_beam = (1/3) * m_beam * L_beam^2.

So, you can calculate the moment of inertia of the system about axis B by plugging in the given values for m_beam and L_beam, and applying the formula for the beam, and adding the moment of inertia of the sphere (I_sphere) as calculated in part a.

Part c:
To determine the constant horizontal force (F) exerted on the brass sphere to produce the given angular acceleration, we need to use the equation for torque (τ) and Newton's second law for rotation.

The torque acting on the system is given by τ = I * α, where τ is the torque, I is the moment of inertia of the system about axis B (calculated in part b), and α is the angular acceleration.

By rearranging this equation, we get F = τ / R, where F is the perpendicular force exerted on the sphere, and R is the radius of the sphere.

So, you can calculate the required force (F) by plugging in the values of I, α, and R.

Part d:
To determine the linear velocity (v) of the brass sphere after the force has been applied for 4.0 seconds, we can use the equation of rotational motion which relates the angular acceleration (α), angular velocity (ω), and time (t): ω = α * t.

Since the system started from rest, the initial angular velocity (ω_initial) is zero. We can use the equation ω = ω_initial + α * t.

Once we have the final angular velocity (ω) of the system after 4.0 seconds, we can relate it to the linear velocity (v) of the brass sphere using the equation v = R * ω, where R is the radius of the sphere.

So, you can calculate the linear velocity (v) of the brass sphere by substituting the values of α, t, and R in the equation v = R * ω.

Remember to always use appropriate units and double-check your calculations.