A cylindrical fishing reel has a mass of
0.899 kg and a radius of 5.21 cm. A friction clutch in the reel exerts a restraining torque of 1.43 N · m if a fish pulls on the line.
The fisherman gets a bite, and the reel begins to spin with an angular acceleration of
59.7 rad/s^2.
What force does the fish exert on the line?
How much line unwinds in 0.869 s?
To determine the force exerted by the fish on the line, we need to consider Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the fisherman's reel is experiencing an angular acceleration, so we need to convert it to linear acceleration. The linear acceleration (a) is related to the angular acceleration (α) and the radius (r) by the equation:
a = α * r
Given that the radius (r) is 5.21 cm, we need to convert it to meters by dividing it by 100 (1 m = 100 cm):
r = 5.21 cm / 100 = 0.0521 m
Now we can calculate the linear acceleration:
a = 59.7 rad/s^2 * 0.0521 m = 3.11 m/s^2
Next, we can calculate the force (F) using Newton's second law:
F = m * a
Given that the mass (m) of the reel is 0.899 kg:
F = 0.899 kg * 3.11 m/s^2 = 2.80 N
Therefore, the force exerted by the fish on the line is 2.80 N.
Now let's calculate how much line unwinds in 0.869 s.
To calculate the distance (d) traveled by the line, we need to use the equation of motion:
d = 1/2 * α * t^2
Given that the angular acceleration (α) is 59.7 rad/s^2 and the time (t) is 0.869 s, we can substitute these values into the equation:
d = 1/2 * 59.7 rad/s^2 * (0.869 s)^2 = 23.8 rad
However, we need to convert the angular displacement (23.8 rad) to linear displacement using the formula:
d = α * r
Substituting the values, we get:
d = 23.8 rad * 0.0521 m = 1.24 m
Therefore, the line unwinds by 1.24 meters in 0.869 seconds.