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 find the angle of the force with respect to the radius in circular motion, we need to understand the concept of centripetal force. In circular motion, an object moves in a circular path due to the presence of a centripetal force acting towards the center of the circle.
In this case, the hammer is being accelerated from rest, and we are given the final speed of the hammer. We can use the formula for centripetal force:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the object
v is the final velocity of the object
r is the radius of the circular motion
First, let's calculate the centripetal force using the given data:
F = (7.30 kg * (21.0 m/s)^2) / 0.80 m
Calculating the numerator:
(7.30 kg * (21.0 m/s)^2) = 3321.3 kg·m^2/s^2
Dividing the numerator by the radius (0.80 m):
F = 3321.3 kg·m^2/s^2 / 0.80 m = 4151.625 kg·m/s^2
Now we have the value for the centripetal force. To find the angle of this force with respect to the radius, we need to find the components of the force along the radius and perpendicular to the radius.
The component of the force along the radius is given by:
F_r = F * cos(θ)
The component of the force perpendicular to the radius is given by:
F_θ = F * sin(θ)
We can rearrange these equations to find the angle θ:
tan(θ) = F_θ / F_r
Let's substitute the values:
tan(θ) = (F * sin(θ)) / (F * cos(θ))
The centripetal force (F) cancels out:
tan(θ) = sin(θ) / cos(θ)
Now we can solve for θ by taking the inverse tangent (arctan) of both sides:
θ = arctan(sin(θ) / cos(θ))
Using a scientific calculator, plug in the values:
θ = arctan(sin(θ) / cos(θ))
The resulting value will give us the angle of the force with respect to the radius of the circular motion.