A hammer thrower accelerates the hammer (mass = 7.30 kg) from rest within four full turns (revolutions) and releases it at a speed of 21.0 m/s. Assuming a uniform rate of increase in angular velocity and a radius of 0.80 m, calculate the angle of this force with respect to the radius of the circular motion

To calculate the angle of the force with respect to the radius of the circular motion, we can use the concept of angular acceleration.

First, let's determine the angular acceleration. Given that the hammer starts from rest and completes four full turns, we can find the initial and final angular velocities.

The number of revolutions is converted to radians by multiplying by 2π:
4 turns * 2π radians/turn = 8π radians

The final angular velocity (ω) can be calculated using the formula:
ω = √(2 * α * θ)
where α is the angular acceleration and θ is the angle covered.

Given that the final speed is 21.0 m/s, we can convert it to angular velocity:
v = r * ω
21.0 m/s = 0.80 m * ω
ω = 21.0 m/s / 0.80 m
ω ≈ 26.25 rad/s

Using the formula mentioned earlier, we can solve for α:
26.25 rad/s = √(2 * α * 8π rad)
26.25 rad/s = √(16π α rad)
26.25^2 rad^2/s^2 = 16π α rad
α = (26.25^2 rad^2/s^2) / (16π rad)
α ≈ 8.13 rad/s^2

Now let's calculate the angle of the force with respect to the radius.

The acceleration of an object moving in a circle is given by a = r * α, where r is the radius and α is the angular acceleration.
In this case, r = 0.80 m and α = 8.13 rad/s^2, so a ≈ 6.50 m/s^2 (rounded to two decimal places).

Using the equation a = F/m, where F is the force and m is the mass of the hammer, we can solve for the force.
F = m * a
F = 7.30 kg * 6.50 m/s^2
F ≈ 47.45 N (rounded to two decimal places)

Finally, to calculate the angle between the force and the radius, we can use the concept of trigonometry.

The force vector F can be split into two components: one parallel to the radius (Fr) and one perpendicular to the radius (Fθ).

The force component Fr is equal to F, and the force component Fθ is equal to F * sin(θ), where θ is the angle.

For equilibrium, the sum of all forces should be equal to zero:
Fr = F * cos(θ)
Fr = 47.45 N * cos(θ)

Since Fr = F, we have:
F = F * cos(θ)
1 = cos(θ)
θ = 0 radians

Therefore, the angle of this force with respect to the radius of the circular motion is 0 radians.