a pendulum 45 cm long swings through a vertical angle of 30 degree.find the height through which the pendulum bob rises.
To find the height through which the pendulum bob rises, you can use the equation:
height = length - length * cosine(angle),
where:
- height is the height through which the pendulum bob rises,
- length is the length of the pendulum, and
- angle is the vertical angle through which the pendulum swings.
Substituting the given values:
length = 45 cm,
angle = 30 degrees,
height = 45 cm - 45 cm * cosine(30 degrees).
To solve this, we need to convert the angle from degrees to radians because the cosine function takes radians as input. 1 degree = π/180.
angle_radians = 30 degrees * (π/180 degrees) = 30 * π/180 radians.
Now, substitute the values and calculate:
height = 45 cm - 45 cm * cosine(30 degrees)
= 45 cm - 45 cm * cosine(30 * π/180 radians).
Using a calculator, compute the cosine of 30 * π/180 radians:
cosine(30 * π/180 radians) = 0.866.
height = 45 cm - 45 cm * (0.866)
= 45 cm - 38.97 cm
≈ 6.03 cm.
Therefore, the height through which the pendulum bob rises is approximately 6.03 cm.
To find the height through which the pendulum bob rises, we can use trigonometry. Here's how you can solve it step by step:
Step 1: Draw a diagram to visualize the situation. Draw a straight vertical line to represent the length of the pendulum, which is 45 cm. From the end of this line, draw a line at an angle of 30 degrees to represent the path of the swinging pendulum bob.
Step 2: From the highest point of the swing, draw a horizontal line to meet the vertical line. This horizontal line represents the height through which the pendulum bob rises.
Step 3: Now, we need to find the length of this horizontal line. We can do this by breaking down the triangle formed by the vertical line, the horizontal line, and the path of the swinging pendulum bob.
Step 4: Since we have a right-angled triangle, we can use trigonometric functions. The sine function relates the opposite side (the height) to the hypotenuse (the length of the pendulum). So, we can use the formula sin(angle) = opposite/hypotenuse.
Step 5: Plug in the value of the angle (30 degrees) and the length of the pendulum (45 cm) into the formula: sin(30) = opposite/45.
Step 6: Solve for the opposite side (which represents the height through which the pendulum bob rises): opposite = sin(30) * 45.
Step 7: Calculate the value of sin(30) using a calculator: sin(30) = 0.5.
Step 8: Substitute the value of sin(30) into the equation: opposite = 0.5 * 45.
Step 9: Calculate the final height through which the pendulum bob rises: opposite = 0.5 * 45 = 22.5 cm.
So, the height through which the pendulum bob rises is 22.5 cm.
Center: A = 0 Deg.
Left of Center: A = 15 Deg.
Right of Center: A = 15 Deg.
h = 45*sin15 = 11.65 cm.