A bug is on the rim of a 78 rev/min, 12 in. diameter record. The record moves from rest to its final angular speed in 3.29 s.
Find the bug’s centripetal acceleration 1.5 s
after the bug starts from rest. (1 in = 2.54
cm).
Answer in units of m/s^2
.
To find the bug's centripetal acceleration 1.5 s after it starts from rest, we first need to find its angular speed at that time.
Given:
Angular speed of the record = 78 rev/min
Diameter of the record = 12 in
Time = 1.5 s
Step 1: Convert the angular speed from rev/min to rad/s:
Angular speed = 78 rev/min = (78 rev/min) * (2π rad/rev) * (1 min/60 s) = 8.167 rad/s
Step 2: Calculate the radius of the record using the diameter:
Radius = 12 in / 2 = 6 in = 6 in * 2.54 cm/in = 15.24 cm = 0.1524 m
Step 3: Determine the angular displacement at 1.5 s using the formula:
θ = ω0 * t + 0.5 * α * t^2
ω0 = 0 (since the record starts from rest)
α = (final angular speed - initial angular speed) / time = (8.167 rad/s - 0 rad/s) / 3.29 s = 2.484 rad/s^2
θ = (0 rad/s) * (1.5 s) + 0.5 * (2.484 rad/s^2) * (1.5 s)^2 = 3.5211 rad
Step 4: Calculate the linear velocity of the bug using the formula:
v = ω * r
v = (8.167 rad/s) * (0.1524 m) = 1.2444 m/s
Step 5: Determine the centripetal acceleration of the bug using the formula:
a = v^2 / r
a = (1.2444 m/s)^2 / (0.1524 m) = 10.1967 m/s^2
Therefore, the bug's centripetal acceleration 1.5 s after it starts from rest is approximately 10.1967 m/s^2.
To find the bug's centripetal acceleration 1.5 s after it starts from rest, we need to follow these steps:
Step 1: Convert the diameter of the record from inches to meters.
Given that 1 inch is equal to 2.54 cm, we can convert the diameter as follows:
12 inches * 2.54 cm/inch * (1 meter / 100 cm) = 0.3048 meters
Step 2: Calculate the radius of the record.
The radius is half the diameter, so:
Radius = 0.3048 meters / 2 = 0.1524 meters
Step 3: Convert the angular speed from revolutions per minute (rev/min) to radians per second (rad/s).
To convert from rev/min to rad/s, we multiply by 2π/60 since there are 2π rad in a revolution and 60 s in a minute.
Angular speed = 78 rev/min * (2π rad/1 rev) * (1 min/60 s) = 8.2 rad/s
Step 4: Calculate the angular acceleration.
Since we know the final angular speed and the time it took to reach that speed, we can use the formula:
Angular acceleration (α) = (final angular speed - initial angular speed) / time = (8.2 rad/s - 0 rad/s) / 3.29 s = 2.497 rad/s^2
Step 5: Calculate the tangential speed of the bug 1.5 s after it starts.
Tangential speed = radius * angular speed = 0.1524 meters * 8.2 rad/s = 1.2508 meters/s
Step 6: Calculate the centripetal acceleration of the bug.
Centripetal acceleration = (tangential speed)^2 / radius = (1.2508 meters/s)^2 / 0.1524 meters = 10.289 m/s^2
Therefore, the bug's centripetal acceleration 1.5 s after it starts from rest is approximately 10.289 m/s^2.
change rev/min to radians/second.
change 12in to meters
centripetal acc=angularspeed^2 * radius
where
angular speed=angularacceler*time