The free-fall acceleration on Mars is 3.7 m/s2.
(a) What length pendulum has a period of 4 s on the earth?
cm
What length pendulum would have a 4 s period on Mars?
cm
(b) A mass is suspended from a spring with force constant 10 N/m. Find the mass suspended from this spring that would result in a 4 s period on Earth and on Mars.
Earth kg
Mars kg
The period of a pendulum is
2*pi*sqrt(L/g)
The period of a spring/mass system is
P = 2*pi*sqrt(m/k)
(k is the spring constant and you can guess what m is)
Use these formulas to solve your problems.
To find the length of a pendulum, you can use the formula:
T = 2π√(L/g)
where T is the period of the pendulum, L is the length of the pendulum, and g is the acceleration due to gravity.
(a) Length of pendulum with a 4 s period on Earth:
On Earth, the acceleration due to gravity is approximately 9.8 m/s^2. So, to find the length of a pendulum with a 4 s period on Earth, we can rearrange the formula:
T = 2π√(L/g)
4 = 2π√(L/9.8)
Simplifying the equation, we get:
2 = π√(L/9.8)
2/π = √(L/9.8)
Squaring both sides, we have:
(2/π)^2 = L/9.8
L = (2/π)^2 * 9.8
Calculating the length, the pendulum on Earth would be approximately 1.0 meters.
(b) Length of pendulum with a 4 s period on Mars:
On Mars, the acceleration due to gravity is given as 3.7 m/s^2. Using the same formula as before:
T = 2π√(L/g)
4 = 2π√(L/3.7)
Simplifying the equation:
2 = π√(L/3.7)
2/π = √(L/3.7)
Squaring both sides:
(2/π)^2 = L/3.7
L = (2/π)^2 * 3.7
Calculating the length, the pendulum on Mars would be approximately 0.3741 meters or 37.41 centimeters.
For part (b), we need to use Hooke's Law for a mass-spring system:
T = 2π√(m/k)
where T is the period, m is the mass, and k is the force constant of the spring.
For Earth:
We want to find the mass that results in a 4 s period on Earth. Using the formula:
T = 2π√(m/k)
4 = 2π√(m/10)
Simplifying:
2 = π√(m/10)
2/π = √(m/10)
Squaring both sides:
(2/π)^2 = m/10
m = 10 * (2/π)^2
Calculating the mass, it would be approximately 1.28 kilograms.
For Mars:
Using the same formula as before:
T = 2π√(m/k)
4 = 2π√(m/10)
Simplifying:
2 = π√(m/10)
2/π = √(m/10)
Squaring both sides:
(2/π)^2 = m/10
m = 10 * (2/π)^2
Calculating the mass, it would still be approximately 1.28 kilograms. The mass suspended from this spring that would result in a 4 s period is the same for both Earth and Mars.