m1 = 2.3 kg block slides on a frictionless horizontal surface and is connected on one side to a spring (k = 45 N/m) as shown in the figure above. The other side is connected to the block m2 = 3.6 kg that hangs vertically. The system starts from rest with the spring unextended.
What is the speed of block m2 when the extension is 65 cm?
Well, if we're talking about the speed of block m2, I guess it's safe to say things are about to get "spring-y"! But hold on tight, because I'm about to "spring" some knowledge on you!
To find the speed of block m2, we need to consider the conservation of energy in the system. When the spring is extended, some potential energy is stored in it. This potential energy gets converted into kinetic energy as the block m2 starts moving.
Given that the extension of the spring is 65 cm, we can calculate the amount of potential energy stored in the spring using the formula:
Potential Energy (PE) = (1/2) * k * x^2
Where k is the spring constant and x is the extension. Plugging in the values, we get:
PE = (1/2) * (45 N/m) * (0.65 m)^2
Now, let's remember the law of conservation of energy. The potential energy stored in the spring gets converted into the kinetic energy of block m2. So:
Potential Energy (PE) = Kinetic Energy (KE)
Now we can find the kinetic energy of block m2:
KE = (1/2) * m2 * v^2
Where m2 is the mass of block m2 and v is its speed. Plugging in the values, we get:
KE = (1/2) * (3.6 kg) * v^2
Now we can equate the potential energy to the kinetic energy:
(1/2) * (45 N/m) * (0.65 m)^2 = (1/2) * (3.6 kg) * v^2
Solving for v, we get:
v^2 = [(45 N/m) * (0.65 m)^2] / (3.6 kg)
v = the square root of {[(45 N/m) * (0.65 m)^2] / (3.6 kg)}
And with some mathematical "spring-ta-tion," you can calculate the answer to find the speed of block m2! I'm sure it will "spring" you some joy when you figure it out!
To determine the speed of block m2 when the extension is 65 cm, we need to analyze the conservation of mechanical energy in the system.
First, let's determine the potential energy stored in the spring when it is extended by 65 cm:
Potential Energy (PE) = (1/2) * k * x^2
where k is the spring constant and x is the extension.
PE = (1/2) * 45 N/m * (0.65 m)^2
= 8.63 J (Joules)
Next, let's find the speed of block m1 when the spring is extended by 65 cm. Since the surface is frictionless and there are no external forces acting on the system, the total mechanical energy is conserved.
Total Mechanical Energy (TME) = Kinetic Energy (KE) + Potential Energy (PE)
Initially, when the system is at rest, the total mechanical energy is zero because there is no kinetic energy (KE) and potential energy (PE). Therefore, the total mechanical energy at any given moment is equal to the potential energy stored in the spring when it is extended by 65 cm.
TME = PE = 8.63 J
Now, let's calculate the kinetic energy of block m1 at the point when it has traveled a distance x (65 cm):
KE = (1/2) * m1 * v^2
where m1 is the mass of block m1 and v is its velocity.
Since the total mechanical energy is conserved, we can equate the kinetic energy to the total mechanical energy:
8.63 J = (1/2) * 2.3 kg * v^2
Simplifying the equation:
v^2 = (2 * 8.63 J) / 2.3 kg
= 7.49 m^2/s^2
Taking the square root of both sides:
v = sqrt(7.49 m^2/s^2)
= 2.74 m/s
Therefore, the speed of block m2 when the extension is 65 cm is 2.74 m/s.
To find the speed of block m2 when the extension of the spring is 65 cm, we can use the principle of conservation of mechanical energy.
First, let's find the potential energy stored in the spring when it is extended by 65 cm. The potential energy stored in a spring can be calculated using the formula:
Potential Energy = (1/2) * k * x^2
Where k is the spring constant (45 N/m in this case) and x is the extension of the spring (65 cm = 0.65 m).
Potential Energy = (1/2) * 45 N/m * (0.65 m)^2
= 8.9125 J
Next, we need to find the total energy of the system at this point. The total energy consists of the potential energy stored in the spring and the kinetic energy of the system. Since the system starts from rest, the total energy is equal to the potential energy.
Total Energy = 8.9125 J
We can equate the total energy to the sum of potential and kinetic energy at any point. The kinetic energy is given by:
Kinetic Energy = (1/2) * m2 * v^2
Where m2 is the mass of block m2 (3.6 kg) and v is the speed of block m2 that we need to find.
Setting the potential energy equal to the kinetic energy and substituting the values, we get:
8.9125 J = (1/2) * 3.6 kg * v^2
Simplifying the equation, we find:
8.9125 J = 1.8 kg * v^2
v^2 = 8.9125 J / 1.8 kg
v^2 = 4.9514 m^2/s^2
Taking the square root of both sides, we find:
v ≈ 2.22 m/s
Therefore, the speed of block m2 when the extension of the spring is 65 cm is approximately 2.22 m/s.
do potential and kinetic energy
k = 45
g = 9.81
x = 0.65 max
Pe = Ke = 0 at start
Pe = (1/2) 45 (.65)^2 - 3.6 (9.81) (.65)
Ke = (1/2)(2.3+3.6) v^2
the sum is zero, solve for v (it could be + and it could be - :)