A drawing shows a device that can be used to measure the speed of a bullet. The device consists of two rotating disks, separated by a distance of d = 0.790 m, and rotating with an angular speed of 65.0 rad/s. The bullet first passes through the left disk and then through the right disk. It is found that the angular displacement between the two bullet holes is θ = 0.360 rad. From these data, determine the speed of the bullet.
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To determine the speed of the bullet, we need to use the concept of conservation of angular momentum.
The angular momentum of an object is given by the equation:
L = Iω
Where L is angular momentum, I is the moment of inertia, and ω is the angular speed.
The moment of inertia for a disk rotating about its axis is given by the equation:
I = (1/2)mr^2
Where m is the mass of the disk and r is the radius.
Since the two disks have the same angular speed, their angular momenta are equal:
L1 = L2
This can be expressed as:
I1ω1 = I2ω2
Since the mass and radius of the disks are the same, the equation becomes:
(1/2)m1r^2ω1 = (1/2)m2r^2ω2
The masses cancel out, and we are left with:
r^2ω1 = r^2ω2
Given that the angular speed of the disks is 65.0 rad/s and the angular displacement between the bullet holes is 0.360 rad, we can setup the equation:
r^2(65.0 rad/s) = r^2ω2 + (0.360 rad)
Simplifying further:
65.0 = ω2 + (0.360/r^2)
We also know that the distance between the disks is 0.790 m. The bullet travels this distance in the time it takes for the disks to rotate through an angle of θ = 0.360 rad.
Using the equation for linear speed, v = rω, we can substitute the values:
0.790 m = r(65.0 rad/s)
Solving for r:
r = 0.790 m / 65.0 rad/s
r ≈ 0.01215 m
Substituting this value back into the equation for ω2:
65.0 = ω2 + (0.360/(0.01215^2))
Simplifying further:
ω2 ≈ 65.0 - 2405.85
ω2 ≈ -2340.85 rad/s (Note: negative sign indicates opposite direction of rotation)
Finally, we can convert the angular speed ω2 to linear speed v2 (bullet speed) using the equation v = rω:
v2 = 0.01215 m * (-2340.85 rad/s)
v2 ≈ -28.49 m/s (Note: negative sign indicates opposite direction of bullet)
Therefore, the speed of the bullet is approximately 28.49 m/s in the opposite direction of the rotation of the disks.
To determine the speed of the bullet, we need to use the concept of angular displacement and the relationship between angular velocity and linear velocity.
The angular displacement, denoted as θ, is the angle through which an object rotates. In this case, θ = 0.360 rad, which is the angular displacement between the two bullet holes.
The angular velocity, denoted as ω, is the rate at which an object rotates. In this case, the disks are rotating at an angular speed of 65.0 rad/s.
The linear velocity, denoted as v, is the speed at which an object moves in a straight line. We want to find the speed of the bullet, which is the linear velocity of the bullet.
The relationship between angular velocity and linear velocity is given by the formula: v = ω * r, where r is the distance from the axis of rotation to the point of interest. In this case, the distance between the two disks is d = 0.790 m.
To find the speed of the bullet, we can solve for v using the given values:
v = ω * r
v = (65.0 rad/s) * (0.790 m)
v = 51.350 m/s
Therefore, the speed of the bullet is approximately 51.350 m/s.
The angular separation of the holes, divided by the angular velocity, is the time it takes the bullet to travel between the plates.
T = 0.360/65 = 5.54*10^-3 s
Divide the plate separation but that time for the bullet velocity. It turns out to be pretty slow, less than the speed of sound. Most bullets are much faster than that.
A similar concept has been used to measure the speed of light.