A simple pendulum of massA simple pendulum of mass 50kg oscillates with an amplitude of 4cm and period of 2s . Determine the angular and linear velocities of the bulb .
Please check for typos. To get angular velocity you need length of the rod.
if the period is 2s, the frequency is f = 1/(2s)
angular velocity ω = 2πf = π rad/s
Now, maximum angle θ/2 occurs at t = 1/4 * 2 = 1/2
so, now we have sin(θ/2) = 4/r
and θ/2 = 1/4 * 2π*1/2 = π/4
4/r = 1/√2
r = 2√2
and so linear velocity is v = rω = 2√2 * π = 2π√2 cm/s
To determine the angular and linear velocities of the bulb in a simple pendulum, we first need to understand the definitions of these quantities.
Angular velocity (ω) is defined as the rate at which the pendulum swings through an angle. It is measured in radians per second (rad/s).
Linear velocity (v) is defined as the rate at which the pendulum moves along its path. It is measured in meters per second (m/s).
Given the following information:
- Mass of the pendulum (m): 50 kg
- Amplitude (A): 4 cm (or 0.04 m)
- Period (T): 2 s
We can use the formula for the period of a pendulum to find the angular velocity:
T = 2π√(L/g)
where L is the length of the pendulum and g is the acceleration due to gravity (approximately 9.8 m/s²).
Since the formula involves the length of the pendulum, we need to convert the given amplitude to the length of the pendulum. In a simple pendulum, the amplitude is half of the maximum displacement, so the length (L) can be calculated as:
L = 2πA
Substituting the given values, we get:
L = 2π(0.04) = 0.2513 m
Plugging this value of L and the acceleration due to gravity (g) into the period formula, we can solve for the angular velocity (ω):
2 = 2π√(0.2513/g)
Squaring both sides, we get:
1 = 4π²(0.2513/g)
Rearranging the equation to solve for g, we get:
g = 4π²(0.2513)
g ≈ 98.372 m/s²
Now, we can substitute this value of g into the period formula to find the angular velocity (ω):
2 = 2π√(0.2513/98.372)
Squaring both sides, we get:
1 = 4π²(0.2513/98.372)
Simplifying the equation, we have:
1 ≈ 0.080
Therefore, the angular velocity (ω) is approximately 0.080 rad/s.
To find the linear velocity (v), we can use the formula:
v = ωL
Substituting the given value of ω and the length of the pendulum (L), we get:
v ≈ 0.080 × 0.2513
v ≈ 0.0201 m/s
Therefore, the linear velocity of the bulb is approximately 0.0201 m/s.