To solve this problem, we need to calculate the maximum number of revolutions per minute (rpm) for which the rope does not snap. The tension in the rope is the centripetal force that keeps the mass moving in a circular motion. The centripetal force is given by the equation:
Fcentripetal=mv2rF_{\text{centripetal}} = \frac{m v^2}{r}
Where:
• mm is the mass of the body (4 kg),
• vv is the tangential velocity of the body,
• rr is the radius of the circular motion (0.4 m, since the rope length is 40 cm).
The maximum tension the rope can withstand is given as 6.4 N, so this is the maximum centripetal force. We will use this maximum tension to calculate the maximum velocity, and then convert that to the number of revolutions per minute.
Step 1: Find the maximum velocity
The maximum tension is equal to the maximum centripetal force, so:
Tmax=FcentripetalT_{\text{max}} = F_{\text{centripetal}} 6.4=mv2r6.4 = \frac{m v^2}{r}
Substitute the known values for mm and rr:
6.4=4⋅v20.46.4 = \frac{4 \cdot v^2}{0.4}
Solving for v2v^2:
6.4×0.4=4⋅v26.4 \times 0.4 = 4 \cdot v^2 2.56=4⋅v22.56 = 4 \cdot v^2 v2=2.564=0.64v^2 = \frac{2.56}{4} = 0.64 v=0.64=0.8 m/sv = \sqrt{0.64} = 0.8 \, \text{m/s}
Step 2: Find the maximum number of revolutions per second
The tangential velocity vv is related to the angular velocity ω\omega by the equation:
v=ω⋅rv = \omega \cdot r
Where ω\omega is the angular velocity in radians per second. Solving for ω\omega:
ω=vr=0.80.4=2 rad/s\omega = \frac{v}{r} = \frac{0.8}{0.4} = 2 \, \text{rad/s}
Now, to find the number of revolutions per second ff, we use the relationship:
f=ω2π=22π=1π rev/sf = \frac{\omega}{2\pi} = \frac{2}{2\pi} = \frac{1}{\pi} \, \text{rev/s}
Step 3: Convert to revolutions per minute
To convert the frequency from revolutions per second to revolutions per minute, multiply by 60:
rpm=60π≈19.1 rev/min\text{rpm} = \frac{60}{\pi} \approx 19.1 \, \text{rev/min}
Final Answer:
The maximum number of revolutions per minute the body can be whirled without snapping the rope is:
19.1\boxed{19.1}
So, the correct answer is B. 19.1.
