Physics homework

Question

A hammer thrower rotates at a rate of 2.9 rev/s just prior to releasing the hammer. If the hammer is located 1.4 m away from the axis of rotation, what is the magnitude of the centripetal acceleration experienced by the athlete (in m/s2)? What is the magnitude of the tension (force) needed to produce this centripetal acceleration if the mass of the hammer is 7.5 kg?

Answer 1

Hi Adriana,

Remember the formula for centripetal acceleration. That formula is a_c = v²/r, where a_c is the centripetal acceleration, v is the tangential velocity, and r is the radius of curvature. For this problem, we are given the radius, but our speed is given to us in revolutions per second. We need to convert revolutions per second to radians per second, and then calculate the tangential velocity from that.

One revolution is 2 pi radians, so 2.9 revolutions per second would be 2.9*2*pi radians per second, or roughly 18.22 radians per second.

To get the tangential velocity t this point, we use the equation v = omega*r, where omega is our radians per second we just calculated. Multiplying 18.22 radians per second by our radius of 1.4m gives us 25.51 m/s.

Now we can plug this into our acceleration formula, a_c = v²/r. Squaring our tangential speed and dividing by the radius gives us a centripetal acceleration of 464.8m/s².

Note - we could have saved a step in our calculations if we remembered that a_c also equals (omega)²*r, but this is not a commonly memorized equation. If we used this, all we would need to do is square 18.22 radians and multiply it by 1.4, which gives the same result of approximately 464.8m/s².

Next, for tension, just remember that tension is a force, and we can find the force of any mass by multiplying it by its net acceleration (or we can get component forces by multiplying the mass by its component accelerations). In this case, the centripetal acceleration is provided by tension in the athlete's arm as an equal but opposite force to the centripetal force required by the hammer to rotate in a circle. Thus, if we calculate the centripetal force of the hammer, we calculate the tension in the athlete's arm.

So F_c=ma_c. The mass of the hammer is 7.5kg and the acceleration, we calculated, is 464.8m/s². So the force is approximately 3,486N. This number is rather large, but the athlete is rotating practically 20lbs of mass in three arm-length circles per second, which is quite impressive, but also takes a lot of force.