A 2.2 kg mass is attached to a spring with a force constant of 560 N/m on a smooth horizontal surface. The mass is pulled to one side, released, and allowed to oscillate back and forth.
(a) find the maximum compression and expansion of the spring if the maximum speed of the mass is 1.7 m/s.
(b) by what multiplicative factor do your answers to part (a) change if the force constant of the spring is doubled?
a)The maximum compression(x)
mg=kx tyen solve for x
x=mg/k =2.2*9.8/560=0.0385
-maximum speed =w=squ of k/m
and Vmax =w*x
.106
Oh, it's time to get springy! Let's dive into these questions like a gymnast on a trampoline.
(a) To find the maximum compression and expansion of the spring, we need to use the relationship between kinetic energy (KE) and potential energy (PE). At maximum speed, all the kinetic energy is converted into potential energy. So, we can write:
KE = PE
0.5 * m * vmax^2 = 0.5 * k * xmax^2
where m is the mass (2.2 kg), vmax is the maximum speed (1.7 m/s), and k is the force constant of the spring (560 N/m). We can solve for xmax:
xmax = sqrt((m * vmax^2) / k)
Plug in the values and calculate xmax. That's the maximum compression and expansion of the spring.
(b) Now, let's double the force constant of the spring (k). This basically means the spring becomes stiffer. To find the new maximum compression and expansion, we need to calculate xmax again, but this time using the new force constant (2k).
xmax_new = sqrt((m * vmax^2) / (2k))
To find the multiplicative factor, we can divide xmax_new by xmax:
Multiplicative factor = xmax_new / xmax
Plug in the values and calculate the factor. And voila, you've got yourself the change when the force constant is doubled!
Remember, the key to enjoying physics is to spring into action and embrace the calculations with a sense of humor. Good luck, my friend!
To find the maximum compression and expansion of the spring, we can use the formula for the maximum speed of an object undergoing simple harmonic motion:
v = Aω
where v is the maximum speed, A is the amplitude (maximum displacement), and ω is the angular frequency.
The angular frequency can be determined using the formula:
ω = √(k/m)
where k is the force constant of the spring and m is the mass.
(a) To find the maximum compression and expansion, we need to determine the amplitude (A). We are given the maximum speed (v = 1.7 m/s) and the mass (m = 2.2 kg).
First, we need to find the angular frequency:
ω = √(k/m) = √(560 N/m / 2.2 kg) = √(254.55 rad/s^2) ≈ 15.96 rad/s
Now, we can find the amplitude:
A = v/ω = 1.7 m/s / 15.96 rad/s ≈ 0.1066 m
Therefore, the maximum compression and expansion of the spring is approximately 0.1066 meters.
(b) If the force constant of the spring is doubled, the new force constant would be 2k.
Using the same formula as before, the new angular frequency can be calculated:
ω' = √((2k)/m) = √((2*560 N/m)/2.2 kg) = √(509.09 rad/s^2) ≈ 22.57 rad/s
Using the formula v = Aω', we can find the new amplitude:
A' = v/ω' = 1.7 m/s / 22.57 rad/s ≈ 0.0754 m
The multiplicative factor by which the answers change is:
A'/A = 0.0754 m / 0.1066 m ≈ 0.7071
Therefore, if the force constant of the spring is doubled, the maximum compression and expansion will reduce to approximately 70.71% of the original value.
To solve this problem, we can use the principle of conservation of mechanical energy. The mechanical energy of the system (spring-mass) is conserved as long as there are no external forces acting on it.
Let's start with part (a) and find the maximum compression and expansion of the spring.
1. Find the potential energy stored in the spring:
The potential energy stored in a spring is given by the formula: U = (1/2)kx^2, where U is the potential energy, k is the force constant of the spring, and x is the displacement from the equilibrium position.
Given k = 560 N/m, we can rearrange the formula to solve for x:
U = (1/2)kx^2
U = (1/2)(560 N/m)x^2
2. Find the maximum kinetic energy of the mass:
The maximum kinetic energy of the mass is achieved when it reaches its maximum speed, which is given as 1.7 m/s.
The kinetic energy of an object is given by the formula: K = (1/2)mv^2, where K is the kinetic energy, m is the mass, and v is the velocity.
Given m = 2.2 kg and v = 1.7 m/s, we can calculate the maximum kinetic energy:
K = (1/2)(2.2 kg)(1.7 m/s)^2
3. Use conservation of mechanical energy:
Since the mechanical energy is conserved, the total mechanical energy at any point in the oscillation is the sum of the potential energy and kinetic energy.
At the maximum compression or expansion point, all the energy is converted to potential energy. Therefore, the potential energy equals the total mechanical energy.
Equating the potential energy to the kinetic energy, we can solve for x:
(1/2)(560 N/m)x^2 = (1/2)(2.2 kg)(1.7 m/s)^2
Simplifying and solving for x, we get:
x^2 = ((2.2 kg)(1.7 m/s)^2) / (560 N/m)
Taking the square root of both sides to find x, we get:
x = sqrt(((2.2 kg)(1.7 m/s)^2) / (560 N/m))
This will give us the maximum compression and expansion of the spring.
For part (b), if the force constant of the spring is doubled (k' = 2k), we can repeat the calculations using the new force constant to find the new value of x:
x' = sqrt(((2.2 kg)(1.7 m/s)^2) / (2 * 560 N/m))
By dividing x' by x, we can calculate the multiplicative factor by which the answer changes.