Suppose a 1.5kg block of wood is slid along a floor and it compresses a spring that is attached horizontally to a wall. The spring constant is 555N/m and the block of wood is traveling 9.0m/s when it hits the spring. Assume that the floor is frictionless and the spring is ideal.
a) By how much does the block of wood compress the spring.
b) If the block of wood attaches to the spring so that the system oscillates back and forth, what will be the amplitude of the oscillation.
KE at impact=PE stored in spring
1/2 m v^2=1/2 k x^2 solve for x
To find the answer to both parts (a) and (b), we need to apply the principle of conservation of mechanical energy.
The conservation of mechanical energy states that the total mechanical energy of a system remains constant if no external forces are acting on it. In this case, since there is no friction and the spring is ideal, we can assume no external forces are acting on the system.
(a) To find how much the block of wood compresses the spring, we need to equate the initial kinetic energy of the block to the potential energy stored in the compressed spring.
The initial kinetic energy (Ek) of the block can be calculated using the formula:
Ek = 0.5 * mass * velocity^2
Given:
mass (m) = 1.5 kg
velocity (v) = 9.0 m/s
Substituting these values into the formula, Ek = 0.5 * 1.5 kg * (9.0 m/s)^2
Next, we can calculate the potential energy stored in the compressed spring using the formula:
Ep = 0.5 * spring constant * distance^2
The spring constant (k) = 555 N/m
Since the block comes to rest at maximum compression, the potential energy stored in the spring is equal to the initial kinetic energy of the block. So we can equate Ek and Ep:
0.5 * 1.5 kg * (9.0 m/s)^2 = 0.5 * 555 N/m * distance^2
Simplifying the equation, we can solve for the distance (compression) the block travels:
distance^2 = (0.5 * 1.5 kg * (9.0 m/s)^2) / (0.5 * 555 N/m)
Now, take the square root of both sides to find the distance:
distance = sqrt((0.5 * 1.5 kg * (9.0 m/s)^2) / (0.5 * 555 N/m))
Evaluate the expression to find the compression distance of the spring.
(b) To find the amplitude of the oscillation, we need to consider that the total energy of the system is conserved between kinetic energy and potential energy. The amplitude represents the maximum displacement from the equilibrium position when the block is oscillating.
Using the conservation of energy, we can equate the initial kinetic energy (Ek) of the block with the maximum potential energy (Ep) of the spring. This occurs when the block is at its maximum displacement (amplitude).
0.5 * 1.5 kg * (9.0 m/s)^2 = 0.5 * 555 N/m * amplitude^2
Solve for the amplitude (distance from the equilibrium position) by rearranging the equation and isolating the amplitude.
amplitude^2 = (0.5 * 1.5 kg * (9.0 m/s)^2) / (0.5 * 555 N/m)
Take the square root of both sides to find the amplitude:
amplitude = sqrt((0.5 * 1.5 kg * (9.0 m/s)^2) / (0.5 * 555 N/m))
Evaluate the expression to find the amplitude of the oscillation.