Question

In an unusual variation on the game of darts, a 530 Part A g, 36-cm-diameter dartboard rotates at 44rpm on a frictionless axle. Two champion dart players simultaneously throw 50 g darts at 7.5 m/s that Afterwards, what is the dartboard's angular velocity? each stick in the dartboard 10 cm from the axle. Express your answer in revolutions per minute. * Incorrect; Try Again; 23 attempts remaining

Answer 1

To find the dartboard's angular velocity after the darts are thrown, we can use the principle of conservation of angular momentum.

Step-by-step explanation:

To find the dartboard's angular velocity after the darts are thrown, we can use the principle of conservation of angular momentum. The total initial angular momentum of the dartboard and the darts is equal to the total final angular momentum. Because the dartboard starts at rest and there are no external torques acting on the system, the initial angular momentum is zero.

The angular momentum of the darts can be calculated by multiplying their moment of inertia by their angular velocity. The moment of inertia of a dart can be approximated as a point mass multiplied by the square of its distance from the axis of rotation. Assuming the darts are thrown perpendicularly to the dartboard, their moment of inertia can be calculated as:

I = mass * distance^2

You can substitute the values given in the question to calculate the moment of inertia of one dart. This value can then be multiplied by the angular velocity of one dart to obtain the total angular momentum of the darts.

To find the final angular velocity of the dartboard, divide the total angular momentum of the darts by the moment of inertia of the dartboard:

angular velocity of dartboard = total angular momentum of darts / moment of inertia of dartboard

Finally, convert the angular velocity to revolutions per minute using the conversion factor: 1 revolution = 2*pi radians.

Answer 2

After the dart hits the dartboard, the angular velocity of the dartboard will be 138 rad/s or 4140 revolutions per minute (rpm).

Step-by-step explanation:

To determine the dartboard's angular velocity after the dart hits it, we can use the principle of conservation of angular momentum. The total angular momentum before the collision is zero since the dartboard is not rotating initially. After the dart hits the dartboard, both the dart and the dartboard will start rotating together. We can use the equation for angular momentum, L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. Since the dart and the dartboard are rotating together, their angular velocities will be equal and opposite in direction.

The moment of inertia of the combined system can be calculated using the equation for the moment of inertia of a rotating object, I = mr^2, where m is the mass of the object and r is the distance of the object from the axis of rotation. In this case, the moment of inertia of the dartboard and the dart will be added since they are rotating together. The dart has a mass of 50g, which is 0.05kg, and it sticks 10cm away from the axis of rotation.

The moment of inertia of the dart is (0.05kg)(0.1m)^2 = 0.0005kgm^2. The dartboard has a mass of 530g, which is 0.53kg, and its radius is 0.36m. The moment of inertia of the dartboard is (0.53kg)(0.36m)^2 = 0.069kgm^2. Therefore, the total moment of inertia is 0.0005kgm^2 + 0.069kgm^2 = 0.0695kgm^2.

Since the dart and the dartboard rotate together, their angular velocities will be the same. Let's assume that the dart and the dartboard rotate with an angular velocity of ω. The angular momentum of the dart can be calculated by multiplying its moment of inertia with its angular velocity, so the angular momentum of the dart is (0.0005kgm^2)ω. The angular momentum of the dartboard can be calculated by multiplying its moment of inertia with its angular velocity, so the angular momentum of the dartboard is (0.069kgm^2)(-ω), since its direction is opposite to that of the dart. Since the total angular momentum before the collision is zero and the sum of angular momentum after the collision is zero, we can set up the equation 0 + (0.0005kgm^2)ω + (0.069kgm^2)(-ω) = 0. Solving for ω, we get ω = 0.069kgm^2 / 0.0005kgm^2 = 138 rad/s.

Finally, to express the answer in revolutions per minute (rpm), we can use the formula 1 revolution = 2π radians. Therefore, 138 rad/s = (138 rad/s)(60s/min)(1 revolution / 2π rad) = 4140 revolutions per minute (rpm).