A block is dropped onto a spring with k = 27 N/m. The block has a speed of 3.2 m/s just before it strikes the spring. If the spring compresses an amount 0.13 m before bringing the block to rest, what is the mass of the block?
To find the mass of the block, we can use the concept of conservation of mechanical energy.
When the block is dropped onto the spring, it has gravitational potential energy due to its height above the ground. This potential energy is converted into the elastic potential energy stored in the compressed spring when the block comes to rest. The initial kinetic energy of the block is also converted into elastic potential energy when it is brought to rest.
The equation for the conservation of mechanical energy is:
Initial potential energy + Initial kinetic energy = Final potential energy + Final kinetic energy
The initial potential energy of the block is given by its gravitational potential energy:
Initial potential energy = m * g * h
Where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height from which the block is dropped.
The initial kinetic energy of the block is given by:
Initial kinetic energy = 0.5 * m * v^2
Where v is the initial speed of the block.
The final potential energy is given by the elastic potential energy stored in the compressed spring:
Final potential energy = 0.5 * k * x^2
Where k is the spring constant and x is the compression of the spring.
The final kinetic energy would be zero, as the block comes to rest.
Using these equations, we can set up the following equation:
m * g * h + 0.5 * m * v^2 = 0.5 * k * x^2
Now, we can substitute the given values:
m * 9.8 * 0 + 0.5 * m * (3.2)^2 = 0.5 * 27 * (0.13)^2
Simplifying further, we get:
0.5 * m * 10.24 = 0.5 * 27 * 0.0169
Now, we can solve for m:
m * 10.24 = 0.5 * 27 * 0.0169
m * 10.24 = 0.5 * 0.4563
m * 10.24 = 0.22815
m = 0.22815 / 10.24
m ≈ 0.02229 kg
Therefore, the mass of the block is approximately 0.02229 kg.