Two identical 1.60 {\rm kg} masses are pressed against opposite ends of a spring of force constant 1.85 {\rm N}/{\rm cm}, compressing the spring by 25.0 {\rm cm} from its normal length.Find the maximum speed of each mass when it has moved free of the spring on a smooth, horizontal lab table.
Well, well, well, looks like these masses are eager to spring into action! Let's see what's bouncing over here.
First, we need to calculate the potential energy stored in the spring when it's compressed. The potential energy stored in a spring is given by the formula:
PE = (1/2)kx²,
where PE is the potential energy, k is the force constant, and x is the displacement.
Plugging in the values, we have:
PE = (1/2)(1.85 N/cm)(25.0 cm)².
Now, we have the potential energy. But hey, why don't we convert those units to SI units just to be consistent?
PE = (1/2)(1.85 N/cm)(0.25 m)².
Simplifying that, we get:
PE = 0.028 Nm.
Now, here's where the fun begins! As the masses move free of the spring, all of that potential energy gets converted into kinetic energy.
Since we have two identical masses, the total kinetic energy of both masses will be twice the kinetic energy of one mass. So, let's find the maximum speed of just one mass.
The kinetic energy is given by the formula:
KE = (1/2)mv²,
where KE is the kinetic energy, m is the mass, and v is the velocity.
Setting the potential energy equal to the kinetic energy:
0.028 Nm = (1/2)(1.6 kg)v².
Solving for v, we get:
v = √(2(0.028 Nm)/(1.6 kg)).
Calculating that, v ≈ 0.23 m/s.
So, the maximum speed of each mass when it has moved free of the spring is approximately 0.23 m/s. They'll be springing away in no time!
To find the maximum speed of each mass when it has moved free of the spring, we can use the principle of conservation of mechanical energy. The maximum potential energy stored in the spring will be converted into kinetic energy of the masses.
1. Calculate the potential energy stored in the spring:
The potential energy stored in the spring can be given by the equation:
Potential energy (PE) = ½ * k * x^2
Where k is the force constant of the spring and x is the compression of the spring.
Given:
Force constant, k = 1.85 N/cm = 1.85 N / 100 cm = 0.0185 N/cm
Compression of the spring, x = 25 cm = 25 cm * 1m/100 cm = 0.25 m
PE = ½ * 0.0185 N/cm * (0.25 m)^2
PE = 0.0185 N/cm * 0.0625 m^2
PE = 0.00115625 N m = 0.00116 J (rounded to four decimal places)
2. Calculate the maximum speed:
The maximum potential energy stored in the spring will be converted into kinetic energy, given by:
Kinetic energy (KE) = ½ * m * v^2
Where m is the mass of each object and v is the maximum speed.
Given:
Mass of each object, m = 1.60 kg
From the conservation of mechanical energy:
Potential energy (PE) = Kinetic energy (KE)
0.00116 J = ½ * 1.60 kg * v^2
v^2 = 0.00116 J / (½ * 1.60 kg)
v^2 = 0.00116 J / 0.8 kg
v^2 = 0.00145 m^2/s^2
Taking the square root of both sides:
v = √(0.00145 m^2/s^2)
v ≈ 0.038 m/s (rounded to three decimal places)
Therefore, the maximum speed of each mass when it has moved free of the spring on a smooth, horizontal lab table is approximately 0.038 m/s.
To find the maximum speed of each mass when it has moved free of the spring, we can use the principle of conservation of mechanical energy. The mechanical energy is conserved when there are no non-conservative forces acting on the system (such as friction or air resistance).
1. First, we need to find the potential energy stored in the spring when it is compressed. The potential energy of a spring can be calculated using the formula:
U = 0.5 * k * x^2
where U is the potential energy, k is the force constant of the spring, and x is the displacement of the spring from its normal length.
In this case, the force constant is given as 1.85 N/cm, and the displacement is 25.0 cm. So we have:
U = 0.5 * (1.85 N/cm) * (25.0 cm)^2
2. Next, we can calculate the maximum speed using the conservation of mechanical energy. The total mechanical energy of the system is the sum of the potential energy stored in the spring and the kinetic energy of the masses when they are free of the spring.
E_total = K + U
where E_total is the total mechanical energy, K is the kinetic energy, and U is the potential energy.
Since the masses are identical, we can find the maximum speed of each mass separately. The kinetic energy can be calculated using the formula:
K = 0.5 * m * v^2
where m is the mass of each mass (1.60 kg) and v is the maximum speed.
Equating the total mechanical energy to the sum of the potential and kinetic energies, we have:
E_total = 0.5 * m * v^2 + U
Rearranging the equation to solve for v, we get:
v^2 = (2 * (E_total - U)) / m
Taking the square root of both sides, we find:
v = sqrt((2 * (E_total - U)) / m)
3. To find the total mechanical energy of the system (E_total), we need to consider that when the masses are free of the spring, their potential energy is zero. Therefore, the total mechanical energy becomes equal to the kinetic energy alone.
E_total = K
4. Substituting the values into the equation, we can calculate the maximum speed:
v = sqrt((2 * K) / m)
v = sqrt((2 * (0.5 * m * v^2)) / m)
Simplifying the equation, we have:
v = sqrt(v^2)
v^2 = v^2
Therefore, any non-zero value of v will satisfy the equation. This means that the maximum speed of each mass is not limited.
In conclusion, the maximum speed of each mass when it has moved free of the spring on a smooth, horizontal lab table is not limited and can be any positive value.
Using conservation of energy. Energy transfers from potential spring to kinetic. 1.85N/cm=185N/m
25cm=.25m
PE=.5*185*(.25^2)=5.78125
PE=KE(final)
KE(final)=1/2(toatl mass)*velocity^2
KE=.5(2*1.6)(v^2)
5.78125=.5*2*1.6*v^2
algebra....v=1.901m/s