1) A circular disk 0.20m in diameter starts from rest and accelerates with constant angular acceleration to an angular acceleration to an angular velocity of 210 rad/s in 10.00s. Find the angular acceleration and the angle which the disk has turned in degrees.
2) A car with tires of radius 32cm drives on the highway at 60km/h. What is the angular speed of the tires.
1. a = (V-Vo)/t = (210-0)/10 =21rad/s^2.
d = 0.5a*t^2 = 0.5*21*10^2 = 1050 Rad.
A = 1050rad * 360o/6.28 = 50,134 Deg..
2. C = pi*2r = 3.14 * 64cm=201 cm=2.01m
= Circumference.
V = 60km/h = 60000m/3600s = 16.67 m/s.
Va = 16.67m/s * 6.28rad/2.01m=52 rad/s.
= Angular velocity.
Step 1:
To find the angular acceleration, we can use the formula:
angular acceleration = (angular velocity - initial angular velocity) / time
Given that the initial angular velocity is 0 rad/s, the final angular velocity is 210 rad/s, and the time is 10.00 s, we can plug these values into the formula:
angular acceleration = (210 rad/s - 0 rad/s) / 10.00 s
Simplifying:
angular acceleration = 210 rad/s / 10.00 s
Step 2:
To find the angle that the disk has turned in degrees, we can use the formula:
angle = initial angular velocity * time + 0.5 * angular acceleration * time^2
Given that the initial angular velocity is 0 rad/s, the angular acceleration is the value we calculated in step 1, and the time is 10.00 s, we can plug these values into the formula:
angle = 0 rad/s * 10.00 s + 0.5 * (210 rad/s / 10.00 s) * (10.00 s)^2
Simplifying:
angle = 0 + 0.5 * 210 rad/s * 100 s
angle = 0 + 0.5 * 21000 rad
Finally, to convert the angle from radians to degrees, we can use the conversion factor:
1 radian = 180 degrees / pi
Therefore:
angle = (0.5 * 21000 rad) * (180 degrees / pi radian)
Simplifying:
angle = 0.5 * 21000 degrees * (180 / pi)
angle ≈ 1909.86 degrees
So, the angular acceleration is approximately 21 rad/s^2 and the angle that the disk has turned is approximately 1909.86 degrees.
To find the answers to these questions, we can use some basic formulas from rotational motion.
1)
Given:
- Diameter of the circular disk = 0.20m
- Initial angular velocity = 0 rad/s
- Final angular velocity = 210 rad/s
- Time taken = 10.00s
To find the angular acceleration, we can use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time taken
Plugging in the values:
α = (210 rad/s - 0 rad/s) / 10.00s
α = 21 rad/s^2
So, the angular acceleration is 21 rad/s^2.
To find the angle which the disk has turned, we can use the formula:
angle (θ) = initial angular velocity * time + (1/2) * angular acceleration * time^2
Plugging in the values:
θ = 0 rad/s * 10.00s + (1/2) * 21 rad/s^2 * (10.00s)^2
θ = 1050 rad
To convert the angle to degrees, we can use the conversion factor:
1 rad = (180/π) degrees
θ (in degrees) = 1050 rad * (180/π) degrees/rad
θ (in degrees) ≈ 60181 degrees
Therefore, the angular acceleration is 21 rad/s^2 and the angle which the disk has turned is approximately 60181 degrees.
2)
Given:
- Radius of the tire (r) = 32cm = 0.32m
- Speed of the car = 60km/h
To find the angular speed of the tire, we can use the formula:
angular speed (ω) = linear speed / radius
First, let's convert the speed of the car from km/h to m/s:
Speed of the car = 60 km/h * (1000 m/1 km) * (1 h/3600 s)
Speed of the car ≈ 16.67 m/s
Now, let's calculate the angular speed:
ω = 16.67 m/s / 0.32 m
ω ≈ 52.09 rad/s
Therefore, the angular speed of the tire is approximately 52.09 rad/s.