A block rests on a frictionless surface. Find its instantaneous acceleration when the spring on the left is compressed 4.48 cm while the spring on the right is stretched 10.3 cm. Each spring has a force constant of 1.05 103 N/m.
To find the instantaneous acceleration of the block, we need to apply Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
Here's how you can calculate the instantaneous acceleration:
1. Determine the net force acting on the block. Since the surface is frictionless, the only forces acting on the block are the forces exerted by the compressed and stretched springs.
2. Calculate the force exerted by each spring using Hooke's law, which states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position.
The force exerted by the left spring (F_left) is given by:
F_left = -k * x_left
Similarly, the force exerted by the right spring (F_right) is given by:
F_right = k * x_right
where k is the force constant of each spring, x_left is the compression distance of the left spring, and x_right is the stretching distance of the right spring.
3. Determine the net force by subtracting the force exerted by the right spring from the force exerted by the left spring:
Net force (F_net) = F_left - F_right
4. Apply Newton's second law: F_net = m * a
where m is the mass of the block and a is the acceleration we're trying to find.
5. Rearrange the equation to solve for the acceleration:
a = F_net / m
6. Substitute the values into the equation:
a = (F_left - F_right) / m
Plug in the values given in the problem:
a = (-k * x_left - k * x_right) / m
Note: The negative sign for the compression distance indicates that the force exerted by the left spring is in the opposite direction.
7. Calculate the value of the acceleration using the given values:
a = (-1.05 * 10^3 N/m * 0.0448 m - 1.05 * 10^3 N/m * 0.103 m) / m
Simplify the equation:
a = (-47.04 N - 108.15 N) / m
a = -155.19 N / m
Therefore, the instantaneous acceleration of the block is -155.19 N/m.