A runner with mass 100 kg jogs around a circular track with circumference 1 km. If he

does one lap in 6 min, what is his angular momentum about the center of the track?

In this case,

Angular Momentum = M*V*R

R = 1000 m/(2 pi) = 159.2 m
V = 1000 m/(360 s) = 2.778 m/s

Now do the MVR multiplication

44000kg

Theradiusofthetrackis R= C =1000m=160m.Thespeedoftherunneris 2π 2π

v = d = 1000m = 2.78 ms . The angular momentum about the center is thus t 360s
L = mvR = (100kg)(2.78 m )(160m) = 44000 Js or 44000 kg m2 . ss

To find the angular momentum of the runner, we need to calculate both the linear momentum and the moment of inertia.

First, let's find the linear momentum of the runner. Linear momentum (p) is given by the formula p = m * v, where m is the mass and v is the velocity.

Given the mass of the runner is 100 kg, we need to determine the velocity. To find the velocity, we divide the circumference of the circular track by the time taken to complete one lap. The circumference of the track is 1 km, which is equal to 1000 meters.

The time taken to complete one lap is given as 6 minutes. However, we need to convert it into seconds since the SI unit for time is seconds. Therefore, 6 minutes will be equal to 6 * 60 = 360 seconds.

Now, we can calculate the velocity of the runner as follows:
velocity = distance / time = circumference / time = 1000 m / 360 s = 2.78 m/s

Next, we can calculate the linear momentum:
linear momentum = mass * velocity = 100 kg * 2.78 m/s = 278 kg·m/s

Now that we have the linear momentum, let's calculate the moment of inertia. The moment of inertia (I) for a circular track can be given by the formula I = m * r^2, where m is the mass and r is the radius of the circular track.

We know the mass of the runner is 100 kg, but we need to determine the radius of the circular track. The radius can be calculated from the circumference using the formula circumference = 2 * π * radius.

Since the circumference is given as 1 km, we can calculate the radius as follows:
circumference = 2 * π * radius
1 km = 2 * 3.14 * radius
radius = 1 km / (2 * 3.14)
radius ≈ 159.2 m

Now we can calculate the moment of inertia:
moment of inertia = mass * radius^2 = 100 kg * (159.2 m)^2 ≈ 2,529,600 kg·m^2

Finally, we can calculate the angular momentum using the formula:
angular momentum = moment of inertia * angular velocity

Since the runner is jogging around the circular track, his angular velocity (ω) can be calculated by dividing 2π radians (one complete rotation) by the time taken to complete one lap:
angular velocity = 2π / time = 2π / 360 s ≈ 0.0175 rad/s

angular momentum = moment of inertia * angular velocity
angular momentum ≈ 2,529,600 kg·m^2 * 0.0175 rad/s

Therefore, the angular momentum of the runner about the center of the track is approximately 44,241 kg·m^2/s.