A uniform rod is set up so that it can rotate about a perpendicular axis at one of its ends. The length and mass of the rod are 0.849 m and 1.17 kg. What constant-magnitude force acting at the other end of the rod perpendicularly both to the rod and to the axis will accelerate the rod from rest to an angular speed of 6.67 rad/s in 9.47 s?

________ N

force*radiusroataion=MomentInertia*alpha

then wf=alpha*time

force=MomentuInertia/radisuroatsion * wf/time

To find the constant-magnitude force acting on the rod, we can use the torque equation: torque = force * distance * sin(θ), where torque is the rotational force, force is the applied force, distance is the distance between the axis of rotation and the point where the force is applied, and θ is the angle between the force vector and the direction in which the force is applied.

In this case, the torque required to accelerate the rod can be expressed as: torque = I * angular acceleration, where I is the moment of inertia of the rod and angular acceleration is the rate at which the angular speed changes.

The moment of inertia of a uniform rod rotating about one end is given by: I = (1/3) * m * L^2, where m is the mass of the rod and L is the length of the rod.

The angular acceleration can be found using the equation: angular acceleration = (final angular speed - initial angular speed) / time.

We are given:
m = 1.17 kg (mass of the rod)
L = 0.849 m (length of the rod)
angular speed = 6.67 rad/s (final angular speed)
time = 9.47 s (time taken)

First, let's calculate the moment of inertia of the rod:
I = (1/3) * m * L^2
= (1/3) * 1.17 kg * (0.849 m)^2

Next, calculate the angular acceleration:
angular acceleration = (final angular speed - initial angular speed) / time
= (6.67 rad/s - 0 rad/s) / 9.47 s

Now, substitute the values into the torque equation:
torque = I * angular acceleration

Finally, given that torque = force * distance * sin(θ), we need to determine the value of distance and sin(θ). In this case, distance is equal to the length of the rod, and sin(θ) can be assumed as 1, since the force is perpendicular to both the rod and the axis.

Therefore, the constant-magnitude force acting on the rod is:
force = torque / (distance * sin(θ))

Now, let's calculate the force:
force = torque / (distance * sin(θ))
= (I * angular acceleration) / (L * 1)

Substitute the calculated values into the equation and solve for force:
force = (I * angular acceleration) / L

By substituting the values into the equation, we can find the force:
force = [(1/3) * 1.17 kg * (0.849 m)^2 * (6.67 rad/s - 0 rad/s) / 9.47 s] / 0.849 m

Calculating the force using a calculator, the result is approximately:
force ≈ 1.93 N

Therefore, the constant-magnitude force acting at the other end of the rod should be approximately 1.93 N to accelerate the rod from rest to an angular speed of 6.67 rad/s in 9.47 seconds.

To find the constant-magnitude force required to accelerate the rod, we can use the equation for rotational motion:

τ = I α

where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

Since the force is acting perpendicularly to the rod and the axis of rotation, the torque can be calculated using the formula:

τ = r × F

where r is the distance between the axis of rotation and the point where the force is applied (which is the length of the rod), and F is the magnitude of the force.

The moment of inertia, I, for a uniform rod rotating about one end is given by:

I = (1/3) * m * L^2

where m is the mass of the rod and L is its length.

We are given the values for the length (L = 0.849 m), mass (m = 1.17 kg), angular speed (ω = 6.67 rad/s), and the time taken (t = 9.47 s).

First, we can calculate the angular acceleration (α) using the formula:

ω = α * t

α = ω / t

α = 6.67 rad/s / 9.47 s

Now, we can substitute the known values into the equation for the moment of inertia:

I = (1/3) * 1.17 kg * (0.849 m)^2

Next, substitute the values for torque (τ) and moment of inertia (I) into the equation for rotational motion:

τ = I * α

r * F = I * α

Finally, solve for the magnitude of the force (F):

F = (I * α) / r

Substitute the known values for I, α, and r into this equation to find the required force.