Circular Motion: Location & Speed at t=1s

[xstetsonx]

1. A computer disk is 8.0 cm in diameter. A reference dot on the edge of the disk is initially located at θ = +45°. The disk accelerates steadily for second, reaching 1480 rpm, then coasts at steady angular velocity for another second. What are the location and speed of the reference dot at t = 1 s?

2. v=rw


3. i already got the 6.28 m/s for speed. and i thought since i have speed i can just do v=rw to get w and then draw a diagram to get the area under the curve. After i got the area just use delta theta/ 2pi=revolution. then minus the whole number and take the decimal to divide by 360 then plus 45 right? but appreantly i am wrong

 

[Andrew Mason]

1. A computer disk is 8.0 cm in diameter. A reference dot on the edge of the disk is initially located at θ = +45°. The disk accelerates steadily for second, reaching 1480 rpm, then coasts at steady angular velocity for another second. What are the location and speed of the reference dot at t = 1 s?

2. v=rw3. i already got the 6.28 m/s for speed. and i thought since i have speed i can just do v=rw to get w and then draw a diagram to get the area under the curve. After i got the area just use delta theta/ 2pi=revolution. then minus the whole number and take the decimal to divide by 360 then plus 45 right? but appreantly i am wrong

We assume it is at rest at t=0 and accelerates until t = 1s at which time its speed $\omege =2960\pi$ rad/sec. To find the angle it covers in that first second, use:

[tex]\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2[/itex]

AM

 

[kerimek]

I would like to clarify some misconceptions in your response. Firstly, the equation v=rw (where v is linear velocity, r is the radius, and w is angular velocity) applies to objects moving in a circular path at a constant speed. In this scenario, the disk is accelerating and then coasting, so this equation cannot be used.

To solve for the location and speed of the reference dot at t=1s, we can use the formula for angular displacement, Δθ = ω0t + 1/2αt^2, where ω0 is the initial angular velocity, α is the angular acceleration, and t is time.

We are given the initial angular velocity (1480 rpm) and the time (1 second), but we need to find the angular acceleration. We can use the formula α = (ωf - ω0)/t, where ωf is the final angular velocity.

Since the disk coasts at a steady angular velocity, ωf = ω0 = 1480 rpm. Converting this to radians per second, we get ωf = 155.5 rad/s.

Plugging in these values, we get α = (155.5 rad/s - 1480 rpm)/1s = -133.5 rad/s^2.

Now, we can plug this value of α into the formula for angular displacement to find the location of the reference dot at t=1s.

Δθ = (1480 rpm)(1 s) + 1/2(-133.5 rad/s^2)(1 s)^2 = 1362.5 radians.

To find the location of the reference dot, we need to convert this to degrees. 1362.5 radians is equivalent to approximately 77699 degrees.

As for the speed, we can use the formula v = rω, where r is the radius (4 cm or 0.04 m) and ω is the angular velocity.

Plugging in the values, we get v = (0.04 m)(155.5 rad/s) = 6.22 m/s. This is slightly different from the value you calculated, but it is because we used the exact value for ωf instead of rounding it to 1480 rpm.

In summary, the location of the reference dot at t=1s is approximately