A 2.9 kg block slides with a speed of 2.1 m/s on a friction less horizontal surface until it encounters a spring. (a) If the block compresses the spring 5.6 cm before coming to rest, what is the force constant on the spring? (b) What initial speed should the block have to compress the spring by 1.4 cm?
the K.E. of the block equals the P.E. of the spring
1/2 m v^2 = 1/2 k x^2
2.9 * 2.1^2 = k * .056^2 ... solve for k ... N/m
2.9 * v^2 = k * .014^2
... substitute k and solve for v
To solve this problem, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy before and after a non-conservative force acts on an object is constant.
(a) To find the force constant, we need to determine the potential energy stored in the spring when it is compressed by 5.6 cm.
1. First, let's convert the compression distance from centimeters to meters:
5.6 cm = 5.6 / 100 m = 0.056 m
2. The potential energy stored in a spring, U_spring, can be calculated using the formula:
U_spring = (1/2) * k * x^2
where k is the force constant of the spring, and x is the compression distance.
Plugging in the values, we have:
U_spring = (1/2) * k * (0.056)^2
3. The initial kinetic energy of the block is equal to the potential energy stored in the spring before it comes to rest. Therefore, we can set the two energies equal to each other:
(1/2) * m * v^2 = (1/2) * k * (0.056)^2
4. Plug in the given values:
(1/2) * 2.9 kg * (2.1 m/s)^2 = (1/2) * k * (0.056 m)^2
5. Solve for k:
k = (2.9 kg * (2.1 m/s)^2) / (0.056 m)^2
k ≈ 2291.58 N/m
So, the force constant on the spring is approximately 2291.58 N/m.
(b) To find the initial speed required to compress the spring by 1.4 cm, we can use the same formula as in part (a), but with the new compression distance.
1. First, convert the compression distance to meters:
1.4 cm = 1.4 / 100 m = 0.014 m
2. Set the initial kinetic energy equal to the potential energy stored in the spring:
(1/2) * m * v^2 = (1/2) * k * (0.014 m)^2
3. Plug in the given values and solve for v:
v = √((k * (0.014 m)^2) / m)
Substituting the known values, we have:
v = √((2291.58 N/m * (0.014 m)^2) / 2.9 kg)
v ≈ 2.453 m/s
So, the initial speed required to compress the spring by 1.4 cm is approximately 2.453 m/s.
To solve this problem, we can use the principle of conservation of mechanical energy, which states that the initial mechanical energy of the system is equal to the final mechanical energy of the system when no non-conservative forces are present (such as friction).
(a) To find the force constant of the spring, we can equate the initial kinetic energy of the block to the potential energy stored in the compressed spring.
1. Start by calculating the initial kinetic energy of the block using the equation:
Kinetic energy (KE) = (1/2) * mass * velocity^2
Substitute the given values:
Mass (m) = 2.9 kg
Velocity (v) = 2.1 m/s
KE = (1/2) * 2.9 kg * (2.1 m/s)^2
2. Next, calculate the potential energy stored in the compressed spring using the equation:
Potential energy (PE) = (1/2) * force constant * displacement^2
Substitute the given values:
Displacement (x) = 5.6 cm = 0.056 m (converted to meters)
Since the block comes to rest, the potential energy equals the initial kinetic energy:
PE = KE
(1/2) * force constant * (0.056 m)^2 = (1/2) * 2.9 kg * (2.1 m/s)^2
Simplify the equation and solve for the force constant:
force constant = (2.9 kg * (2.1 m/s)^2) / (0.056 m)^2
Calculate the value to find the force constant.
(b) To find the initial speed needed to compress the spring by 1.4 cm, we can repeat the steps above using the given displacement value.
1. Start by calculating the potential energy stored in the compressed spring (PE) using the equation:
PE = (1/2) * force constant * displacement^2
Substitute the given values:
Displacement (x) = 1.4 cm = 0.014 m (converted to meters)
Force constant (k) = the value obtained in part (a)
PE = (1/2) * force constant * (0.014 m)^2
2. Find the initial kinetic energy (KE) by equating it to the potential energy stored in the compressed spring:
KE = PE
(1/2) * mass * velocity^2 = (1/2) * force constant * (0.014 m)^2
Solve for the initial velocity (v) by rearranging the equation and taking the square root:
velocity = √((force constant * (0.014 m)^2) / mass)
Calculate the value to find the initial velocity.
By following these steps, you should be able to determine the force constant of the spring in part (a) and the initial velocity required to compress the spring by 1.4 cm in part (b).
(a) Well, I have to admit, this block really knows how to spring into action! To find the force constant on the spring, we can use Hooke's Law. Hooke's Law states that the force exerted by a spring is directly proportional to its displacement. So, we have:
F = kx
Where F is the force, k is the force constant, and x is the displacement. In this case, the displacement is 5.6 cm, which is equivalent to 0.056 meters. The force is zero when the block comes to rest, so we can rewrite Hooke's Law as:
0 = k * 0.056
Solving for k, we find:
k = 0.
So, the force constant on the spring is zero. Looks like our block didn't find a very springy spring after all! Maybe it should try searching on Amazon.
(b) Now, the block wants to compress the spring by 1.4 cm. Let's calculate the required initial speed for this bouncing adventure. To do that, we can use the principle of conservation of mechanical energy.
The initial mechanical energy of the block is equal to the potential energy stored in the spring when it's compressed. The potential energy stored in a spring is given by:
PE = (1/2)kx^2
Where PE is the potential energy, k is the force constant, and x is the displacement. We can set the initial kinetic energy of the block equal to the potential energy of the spring:
(1/2)m*v^2 = (1/2)k*x^2
Plugging in the values, we have:
(1/2)*(2.9)*(v^2) = (1/2)*(k)*((0.014)^2)
Simplifying this equation, we find:
v^2 = (k/m)*((0.014)^2)
Now, we need to solve for v. Unfortunately, we can't determine the exact value for v or k. We need more information, such as the mass of the block or the value of the force constant. So, without this extra info, I'm afraid I can't help you find the magical initial speed the block needs. I guess we'll just have to let it figure that one out on its own!