(part 1)
An athlete whirls an 8.1 kg hammer tied to the end of a 1.4 m chain in a horizontal circle. The hammer moves at the rate of 0.597 rev/s.
What is the centripetal acceleration of the hammer? Assume his arm length is included in the length given for the chain.
Answer in units of m/s2
(part 2)
What is the tension in the chain? Answer in units of N
To find the centripetal acceleration of the hammer, you can use the formula:
ac = (v^2) / r
Where:
ac is the centripetal acceleration,
v is the linear speed of the hammer, and
r is the radius of the circular path.
In this case, the linear speed of the hammer is given as 0.597 rev/s. To convert this to linear speed, we need to multiply it by 2π and the radius of the circular path.
Given that the length of the chain is 1.4 m and the athlete's arm is included in this length, the radius would be the chain length minus the arm length.
So, the radius (r) would be 1.4 m - 0.8 m (assuming a typical arm length of 0.3 m), which equals 0.6 m.
Let's calculate the linear speed (v) first:
v = (0.597 rev/s) * (2π * 0.6 m/rev)
v ≈ 2.26 m/s
Now we can calculate the centripetal acceleration (ac):
ac = (2.26 m/s)^2 / 0.6 m
ac ≈ 8.54 m/s^2
Therefore, the centripetal acceleration of the hammer is approximately 8.54 m/s^2.
Now, moving on to the tension in the chain:
To find the tension, we need to consider the forces acting on the hammer. There are two forces here: the tension in the chain (T) and the weight of the hammer (mg).
The tension in the chain provides the centripetal force required to keep the hammer moving in a circular path. So, we can equate the tension to the centripetal force:
T = (m * ac)
Where:
T is the tension in the chain,
m is the mass of the hammer, and
ac is the centripetal acceleration.
Given that the mass of the hammer is 8.1 kg and the centripetal acceleration is 8.54 m/s^2, we can calculate the tension:
T = (8.1 kg) * (8.54 m/s^2)
T ≈ 69.2 N
Therefore, the tension in the chain is approximately 69.2 N.