A 50.0g Bullet traveling horizontally at 200.0 m/s embeds itself in a much more massive wooden block initially at rest on a horizontal surface (μ=0.10) The block then slides 1.2m toward an ideal spring and collides with it. The block compresses the spring (k=600.0 N/m) a maximum of 20.0 cm. Calculate the mass of the block of wood.
720kg
To find the mass of the wooden block, we need to use the principles of conservation of momentum and conservation of mechanical energy.
Step 1: Find the initial velocity of the bullet-block system.
Given that the bullet has a mass of 50.0 g (0.050 kg) and a velocity of 200.0 m/s, we can calculate its initial momentum.
Momentum (p) = mass (m) x velocity (v)
Initial momentum of the bullet (p_bullet) = (0.050 kg) x (200.0 m/s)
Step 2: Find the final velocity of the bullet-block system.
Since the bullet embeds itself in the wooden block, the final velocity of the system is the same as the velocity of the block after the collision.
Step 3: Find the velocity and momentum of the block after the collision.
Using the principle of conservation of momentum, we can equate the initial momentum of the bullet to the final momentum of the bullet-block system.
Initial momentum of the bullet = Final momentum of the bullet-block system
(p_bullet) = (mass bullet + mass block) x velocity_block
Step 4: Solve for the mass of the block.
We need to rearrange the equation to isolate the mass of the block.
mass_block = (p_bullet) / (velocity_block) - mass_bullet
Step 5: Calculate the work done by the block to compress the spring.
The work done by an object is given by the formula:
Work = (1/2) x k x x^2
where k is the spring constant (600.0 N/m) and x is the displacement of the block (0.20 m).
Step 6: Find the work done by the friction.
The work done by friction can be calculated using:
Work_friction = friction force x distance
The friction force is given by the equation:
Friction force = μ x normal force
where μ is the coefficient of friction (0.10) and the normal force is equal to the weight of the block.
Step 7: Combine the work done by the block and the work done by friction.
Using the principle of conservation of mechanical energy, we can equate the work done by the block to compress the spring to the work done by friction.
Work by block = Work_friction
Step 8: Solve for the mass of the block.
From the equation in Step 7, we can rearrange it to calculate the mass of the block.
mass_block = (2 x Work_friction) / (k x x^2)
Now, we can apply these steps to solve the problem.