A 4.0-kg block and a 2.0-kg block are connected to opposite ends of a relaxed spring of spring constant 300 N/m. The blocks are pushed toward each other for 0.5 s. The 4.0-kg block is pushed with a force of 53 N , and the 2.0-kg block is pushed with a force of 20 N. Ignore the inertia of the spring.
To determine the displacement of the blocks and the potential energy stored in the spring, we need to follow these steps:
1. Calculate the acceleration of each block. We can use Newton's second law of motion (F = ma) to find the acceleration of each block. For the 4.0 kg block, the force applied is 53 N, so the acceleration can be calculated using the formula a = F/m = 53 N / 4.0 kg. Similarly, for the 2.0 kg block, the force applied is 20 N, so the acceleration is a = 20 N / 2.0 kg.
2. Determine the net force acting on the system. Since the blocks are connected by the spring, the spring exerts an equal and opposite force on each block. Therefore, the net force acting on the system is the difference between the force applied to the 4.0 kg block and the force applied to the 2.0 kg block.
3. Calculate the displacement of the blocks. Using the equation of motion s = ut + 0.5at^2, where s is the displacement, u is the initial velocity (which is 0 as the blocks start from rest), a is the acceleration, and t is the time (0.5 s), we can find the displacement for both blocks.
4. Calculate the potential energy stored in the spring. Using the formula for potential energy of a spring, U = 0.5kx^2, where U is the potential energy, k is the spring constant (300 N/m), and x is the displacement, we can find the potential energy stored in the spring.
Let's apply these steps to calculate the displacement of the blocks and the potential energy stored in the spring:
1. Acceleration of the 4.0 kg block:
a = F / m = 53 N / 4.0 kg = 13.25 m/s^2
Acceleration of the 2.0 kg block:
a = F / m = 20 N / 2.0 kg = 10.0 m/s^2
2. Net force on the system:
Net force = Force on the 4.0 kg block - Force on the 2.0 kg block
= 53 N - 20 N
= 33 N
3. Displacement of the blocks:
Displacement (s) = ut + 0.5at^2
For the 4.0 kg block:
s = 0 + 0.5 * (13.25 m/s^2) * (0.5 s)^2
s = 0.5 * 13.25 m/s^2 * 0.25 s^2
s = 0.5 * 13.25 m * 0.0625 s^2
s = 0.4125 m
For the 2.0 kg block:
s = 0 + 0.5 * (10.0 m/s^2) * (0.5 s)^2
s = 0.5 * 10.0 m/s^2 * 0.25 s^2
s = 0.5 * 10.0 m * 0.0625 s^2
s = 0.3125 m
4. Potential energy stored in the spring:
U = 0.5kx^2
For the 4.0 kg block:
U = 0.5 * 300 N/m * (0.4125 m)^2
U = 0.5 * 300 N/m * 0.17015625 m^2
U = 25.51875 J
For the 2.0 kg block:
U = 0.5 * 300 N/m * (0.3125 m)^2
U = 0.5 * 300 N/m * 0.09765625 m^2
U = 14.6484375 J
Therefore, the displacement of the 4.0 kg block is 0.4125 m, and the displacement of the 2.0 kg block is 0.3125 m. The potential energy stored in the spring is 25.51875 J for the 4.0 kg block and 14.6484375 J for the 2.0 kg block.