To test the speed of a bullet, you create a pendulum by attaching a 5.40 kg wooden block to the bottom of a 2.40 m long, 0.700 kg rod. The top of the rod is attached to a frictionless axle and is free to rotate about that point.
You fire a 10 g bullet into the block, where it sticks, and the pendulum swings out to an angle of 32.5°. What was the speed of the bullet?
To find the speed of the bullet, we can use the principle of conservation of mechanical energy.
1. The initial mechanical energy of the system is equal to the final mechanical energy.
Initial mechanical energy = kinetic energy of the bullet
2. The final mechanical energy of the system is the sum of the potential energy and the kinetic energy of the pendulum. Since the pendulum reaches its maximum height at an angle of 32.5°, the potential energy can be calculated using:
Final potential energy = mgh
where m = mass of the system (bullet + wooden block), g = acceleration due to gravity, and h = maximum height reached by the pendulum.
3. The bullet's kinetic energy can be calculated using:
Initial kinetic energy = (1/2)mv^2
where m = mass of the bullet and v = velocity of the bullet.
4. Equate both the initial and final mechanical energies:
(1/2)m_bulletv_bullet^2 = m_totsystemgh
5. Rearrange the formula to solve for the speed of the bullet, v_bullet:
v_bullet = sqrt(2gh)
Now let's calculate the speed of the bullet step by step.
1. Convert the given mass of the bullet from grams to kilograms:
Mass of the bullet = 10 g = 0.01 kg
2. Calculate the total mass of the system:
Total mass of the system = mass of the bullet + mass of the wooden block = 0.01 kg + 5.4 kg = 5.41 kg
3. Acceleration due to gravity, g = 9.8 m/s^2
4. Calculate the maximum height reached by the pendulum:
Maximum height, h = (length of the rod) - (length of the pendulum at the maximum angle)
h = 2.4 m - [(2.4 m) * cos(32.5°)]
h = 2.4 m - (2.4 m * 0.848)
h = 2.4 m - 2.0352 m
h = 0.3648 m
5. Calculate the speed of the bullet:
v_bullet = sqrt(2 * 9.8 m/s^2 * 0.3648 m)
v_bullet = sqrt(7.14864)
v_bullet ≈ 2.673 m/s
Therefore, the speed of the bullet is approximately 2.673 m/s.
To determine the speed of the bullet, we can apply the principles of conservation of momentum and conservation of mechanical energy.
First, let's break down the problem and identify the relevant quantities:
Given:
- Mass of the wooden block (m1) = 5.40 kg
- Length of the rod (L) = 2.40 m
- Mass of the rod (m2) = 0.700 kg
- Angle of swing (θ) = 32.5°
- Mass of the bullet (m_bullet) = 10 g = 0.010 kg
We are asked to find the speed of the bullet (v_bullet).
To solve this problem, we'll follow these steps:
1. Calculate the initial angular velocity of the pendulum.
2. Use the initial angular velocity to find the initial linear velocity of the block.
3. Apply the conservation of momentum to find the velocity of the bullet-block system.
4. Calculate the speed of the bullet.
Now, let's dive into the calculations:
Step 1: Calculate the initial angular velocity of the pendulum.
The initial angular velocity (ω_initial) can be found using the relationship between linear and angular velocity:
ω_initial = v_block_initial / L
where v_block_initial is the initial linear velocity of the block (which we'll find in step 2).
Step 2: Use the initial angular velocity to find the initial linear velocity of the block.
The initial linear velocity of the block (v_block_initial) can be calculated using the equation for the rotational kinetic energy of the pendulum:
0.5 * I_total * ω_initial^2 = 0.5 * m2 * v_block_initial^2
where I_total is the moment of inertia of the system, which can be calculated as:
I_total = I_rod + I_block
and
I_rod = (1/3) * m2 * L^2 (moment of inertia of the rod)
I_block = m1 * L^2 (moment of inertia of the block)
Step 3: Apply the conservation of momentum to find the velocity of the bullet-block system.
Before the collision, the total momentum is given by:
p_initial = m_bullet * v_bullet
After the collision, the block and bullet move as one object.
The total momentum after the collision is given by:
p_final = (m_bullet + m1) * v_final
where v_final is the final velocity of the bullet-block system.
Since momentum is conserved, we can equate the expressions for p_initial and p_final:
m_bullet * v_bullet = (m_bullet + m1) * v_final
Step 4: Calculate the speed of the bullet.
Now, we can solve the equation obtained in step 3 for v_bullet:
v_bullet = (m_bullet + m1) * v_final / m_bullet
By following these steps and plugging in the given values, you can calculate the speed of the bullet.