A turntable rotates counterclockwise at 63 rpm. A speck of dust on the turntable is at θ = 0.57 rad at t = 0 s. What is the angle of the speck at t = 9.5 s? Your answer should be between 0 and 2π rad.
Va=63rev/min * 6.28rad/rev * 1min/60s=
6.594 rad/s.
Angle=0.57 + 6.594rad/s*9.5s=63.21 rad.
or 0.4132 rad.
Well, let's do some spinning math, shall we?
First, we need to find out how many revolutions the speck completes in 9.5 seconds.
Since the turntable rotates counterclockwise at 63 rpm (revolutions per minute), the turntable completes 63/60 = 1.05 revolutions in one second.
Therefore, in 9.5 seconds, the turntable completes 9.5 * 1.05 = 9.975 revolutions.
Now, since the speck of dust is initially at θ = 0.57 rad, and each revolution corresponds to an angle of 2π rad, we can find the angle of the speck at t = 9.5 s by adding the initial angle to the angle covered by the revolutions.
The initial angle of 0.57 rad plus the angle covered by 9.975 revolutions (which is 9.975 * 2π rad) gives us:
0.57 + (9.975 * 2π) ≈ 20π + 0.57 rad.
Since 20π is approximately 62.83, we can say that the angle of the speck at t = 9.5 s is:
62.83 + 0.57 ≈ 63.4 radians.
So, the angle of the speck at t = 9.5 s is approximately 63.4 radians. See, spinning calculations can be quite revolting!
To find the angle of the speck at t=9.5s, we can use the formula:
θ = θ₀ + ωt
where θ is the angle at time t, θ₀ is the initial angle, ω is the angular velocity (in radians per second), and t is the time elapsed.
Given that the turntable rotates counterclockwise at 63 rpm, we can convert the angular velocity to radians per second:
ω = (63 rpm) * (2π rad/1 min) * (1 min/60 s) = 6.3π rad/s
Substituting the given values into the formula, we have:
θ = 0.57 rad + (6.3π rad/s) * (9.5 s)
Calculating the above expression, we find:
θ ≈ 0.57 rad + (59.85π rad) ≈ 188.524 rad
Now, since we want the angle between 0 and 2π rad, we can use the fact that one full revolution is equal to 2π radians. Therefore, we can subtract 2π from the angle until we get a value within the range:
θ ≈ 188.524 rad - (2π rad) ≈ 179.258 rad
Therefore, the angle of the speck at t = 9.5 s is approximately 179.258 radians.
To find the angle of the speck at t = 9.5 s, we need to know the angular displacement of the turntable.
The angular displacement (θ) is given by the formula: θ = ω * t,
Where:
- θ represents the angular displacement in radians
- ω represents the angular velocity in radians per second
- t represents the time in seconds.
Given that the turntable rotates counterclockwise at 63 rpm, we first need to convert the angular velocity to radians per second.
We know that 1 revolution equals 2π radians, and there are 60 seconds in a minute. Thus, the conversion factor is:
1 revolution = 2π radians = 2π/60 radians per second
Therefore, the angular velocity ω can be calculated as:
ω = (63 rpm) * (2π/60) radians per second
Let's calculate the value of ω:
ω = (63 rpm) * (2π/60) = (63 * 2π) / 60 ≈ 6.597 radians per second
Now that we know the angular velocity, we can calculate the angular displacement at t = 9.5 s:
θ = ω * t = 6.597 radians per second * 9.5 s
Let's calculate the value of θ:
θ = 6.597 radians per second * 9.5 s ≈ 62.619 radians
So, the angle of the speck at t = 9.5 s is approximately 62.619 radians.
Please note that the answer should be between 0 and 2π radians, so we need to find the equivalent angle within this range.
To do that, we can divide 62.619 radians by 2π and take the remainder:
Equivalent angle = 62.619 radians mod 2π
Let's calculate the equivalent angle:
Equivalent angle = 62.619 radians mod 2π ≈ 5.024 radians
Therefore, the angle of the speck at t = 9.5 s is approximately 5.024 radians.