A simple pendulum has a period of 4.50s when the length of the pendulum is shortened by 1.2m, the period is 2.50s. calculate the original length of the pendulum.
Original length of the pendulum = (4.50s/2.50s) x 1.2m = 2.4m
A body of mass 20g is made to perform simple harmonic motion at a frequency of 7Hz. Its speed when at a distance of 10cm from the mean position is 220cm/s. Calculate its 1) period ,2) angular speed,3) amplitude
To solve this problem, we can use the equation for the period of a simple pendulum:
T = 2π√(L/g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
Let's consider two scenarios: the original length of the pendulum (L) and the shortened length (L - 1.2).
1. Original period: T1 = 4.50s
2. Shortened period: T2 = 2.50s
Now, we can set up two equations using the above formula:
T1 = 2π√(L/g) --------(1)
T2 = 2π√((L - 1.2)/g) --------(2)
Dividing equation (2) by equation (1), we get:
(T2/T1) = √((L - 1.2)/L)
Squaring both sides of the equation, we have:
(T2/T1)^2 = (L - 1.2)/L
Cross-multiplying, we get:
L(T2/T1)^2 = L - 1.2
Simplifying, we get:
L - L(T2/T1)^2 = 1.2
Factoring out L on the left side, we have:
L (1 - (T2/T1)^2) = 1.2
Dividing both sides by (1 - (T2/T1)^2), we get:
L = 1.2 / (1 - (T2/T1)^2)
Now, substitute the given values:
T1 = 4.50s
T2 = 2.50s
L = 1.2 / (1 - (2.50/4.50)^2)
L = 1.2 / (1 - 0.308)
L = 1.2 / 0.692
L ≈ 1.73 meters
Therefore, the original length of the pendulum is approximately 1.73 meters.
To solve this problem, let's use the formula for the period of a simple pendulum:
Period (T) = 2π√(L/g)
Where:
T = Period
L = Length of the pendulum
g = Acceleration due to gravity (approximately 9.8 m/s²)
Step 1: Determine the original period:
In the given problem, when the length of the pendulum is shortened by 1.2m, the period becomes 2.50s. Let's call the original period T₁ and the shortened period T₂. From the information given, we have:
T₁ = 4.50s
T₂ = 2.50s
Step 2: Solve for the original length (L₁):
Using the formula for the period, we can set up two equations based on the information given:
T₁ = 2π√(L₁/g)
T₂ = 2π√((L₁ - 1.2)/g)
Step 3: Solve the equations:
First, divide both equations by 2π to isolate the square root:
T₁/(2π) = √(L₁/g)
T₂/(2π) = √((L₁ - 1.2)/g)
Next, square both sides of the equations to eliminate the square root:
(T₁/(2π))² = L₁/g
(T₂/(2π))² = (L₁ - 1.2)/g
Step 4: Substitute the values for T₁, T₂, and g:
Substitute T₁ = 4.50s, T₂ = 2.50s, and g = 9.8 m/s² into the equations:
(4.50/(2π))² = L₁/9.8
(2.50/(2π))² = (L₁ - 1.2)/9.8
Simplify the equations:
(1.43029089)² = L₁/9.8
(0.812297331)² = (L₁ - 1.2)/9.8
Step 5: Solve for L₁ by cross-multiplying:
(1.43029089)² × 9.8 = L₁
(0.812297331)² × 9.8 + 1.2 = L₁
L₁ ≈ 20.8457 m (rounded to four decimal places)
Therefore, the original length of the pendulum was approximately 20.8457 meters.