an athlete whirls a 7kg hammer tied to the end of a 1.3m chain
It's asking for the centripetal acceleration of the hammer and the tension of the chain.
So what? There is no question here.
To answer this question, we need to calculate the angular velocity of the hammer. The angular velocity (ω) is a measure of how fast an object rotates around a given axis.
The formula to calculate angular velocity is:
ω = v / r
Where:
ω = angular velocity (in radians per second)
v = linear velocity (in meters per second)
r = radius (in meters)
Given that the hammer is tied to the end of a chain and whirls around, the linear velocity is equal to the speed of the hammer (v = speed).
Next, we can calculate the speed of the hammer using the formula for centripetal force:
F = m * r * ω^2
Where:
F = centripetal force (in newtons)
m = mass of the hammer (in kilograms)
r = radius (in meters)
ω = angular velocity (in radians per second)
In this case, the centripetal force can be approximated as the weight of the hammer, since it is being rotated in a horizontal plane. The weight can be calculated using the formula:
F = m * g
Where:
F = force (in newtons)
m = mass (in kilograms)
g = acceleration due to gravity (approximately 9.8 m/s^2)
Since the hammer is being whirled, the force (F) acting on it is equal to the tension in the chain. So we can equate the equations for centripetal force and weight to find the angular velocity.
Setting the equations equal to each other:
m * g = m * r * ω^2
Simplifying:
g = r * ω^2
Rearranging the equation:
ω^2 = g / r
Solving for ω:
ω = sqrt(g / r)
Substituting the given values:
m = 7 kg
r = 1.3 m
g = 9.8 m/s^2
Plugging in the values:
ω = sqrt(9.8 / 1.3)
ω ≈ 2.796 rad/s
Therefore, the angular velocity of the hammer is approximately 2.796 radians per second.