A pendulum swings through an angle of 14 degrees each second. If the pendulum is 17cm in length and the complete swing from right to left lasts 3 seconds, what area is covered by each complete swing?
Well, that's a swinging question! Let's calculate the area covered by each complete swing of the pendulum.
To find the area covered, we first need to calculate the arc length of the swing. The arc length of a pendulum is given by the formula:
Arc Length = Length of Pendulum × Angle (in radians)
Since the angle is given in degrees, we need to convert it to radians. Luckily, there are 2π radians in a complete circle (360 degrees). So, we can convert 14 degrees to radians by using the following conversion:
Angle (in radians) = Angle (in degrees) × π / 180
Using this conversion, the angle in radians becomes:
Angle (in radians) = 14 degrees × π / 180 ≈ 0.2443461 radians
Now, we can calculate the arc length:
Arc Length = 17 cm × 0.2443461 radians ≈ 4.1518887 cm
Since each complete swing consists of a back-and-forth motion, the area covered is twice the arc length. Therefore, the area covered by each complete swing is approximately:
Area = 2 × 4.1518887 cm ≈ 8.3037774 cm²
So, each complete swing of the pendulum covers an area of approximately 8.3037774 square centimeters. Bet you didn't know pendulums could swing that far and wide!
To find the area covered by each complete swing of the pendulum, we need to calculate the length of the arc traced by the pendulum.
The length of the arc can be calculated using the formula:
Length of arc = radius * angle in radians
First, we need to convert the angle from degrees to radians. Since there are 360 degrees in a circle and 2π radians in a circle, we can use the following conversion formula:
1 radian = (π/180) degrees
Angle in radians = (Angle in degrees) * (π/180)
Angle in radians = 14 degrees * (π/180) ≈ 0.2443 radians
Next, we calculate the length of the arc:
Length of arc = radius * angle in radians
Length of arc = 17 cm * 0.2443 ≈ 4.15 cm
Since each complete swing consists of two arcs (right to left and left to right), the total length covered by each complete swing is:
Total length covered = 2 * Length of arc
Total length covered = 2 * 4.15 cm ≈ 8.3 cm
Now, to find the area covered by each complete swing, we can use the formula for the area of a circle sector:
Area = (angle in radians / 2π) * π * (radius)^2
Area = (0.2443 / (2π)) * π * (17 cm)^2
Area ≈ 4.65 cm^2
Therefore, each complete swing of the pendulum covers an area of approximately 4.65 square centimeters.
To find the area covered by each complete swing of the pendulum, we can use the formula for the area of a sector of a circle.
The formula for the area of a sector is given by:
Area = (θ/360) * π * r^2
where θ is the angle in degrees, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14159.
In this case, the angle covered by each swing of the pendulum is 14 degrees, and the length of the string (radius of the circular swing) is 17 cm.
First, let's convert the angle from degrees to radians. Since there are 360 degrees in a full circle and 2π radians in a full circle, we can use the following conversion factor:
Radians = (π/180) * Degrees
Therefore, the angle in radians is:
θ = (π/180) * 14
Now we can calculate the area:
Area = (θ/360) * π * r^2 = ((π/180) * 14/360) * π * (17)^2
Let's calculate the numerical value of the area.