Final answer:
To find the maximum speed the masses can have without pulling the screws, we calculate the centripetal force using F = m × (v^2/r) with the given maximum force, mass, and the bar's radius. Solving for v, the maximum speed is approximately 3.61 m/s.
Step-by-step explanation:
To determine the maximum speed the masses can have without pulling the screws, we must consider the force produced by the rotating masses at the ends of the bar due to centripetal acceleration. The formula for this force is:
F = m × a
where F is the force, m is the mass of the object, and a is the centripetal acceleration. Since we are dealing with a circular motion, the centripetal acceleration can be expressed as × a = v^2/r, where v is the linear velocity, and r is the radius of the motion.
Now, given that each screw can support a maximum force of 75.0 N and the mass m is 1.15 kg, we can set up the following equation with the maximum force F_max:
F_max = m × (v^2/r)
The length of the bar is 40.0 cm, so the radius r is half of that, which is 20.0 cm or 0.20 m. Substituting F_max and r into the equation, we can solve for v:
75.0 N = 1.15 kg × (v^2/0.20 m)
Multiplying both sides by 0.20 m and then dividing by 1.15 kg, we get:
v^2 = (75.0 N × 0.20 m) / 1.15 kg
To find v, we take the square root:
v = sqrt((75.0 N × 0.20 m) / 1.15 kg)
Upon calculation:
v ≈ sqrt((15 N×m) / 1.15 kg)
Finally, we find that the maximum speed v ≈ sqrt(13.0435 m^2/s^2), which gives a v ≈ 3.61 m/s.
This is the maximum speed at which the bar can rotate without exceeding the maximum force the screws can support.