A 25 rifle bullet traveling 280 buries itself in a 3.8 pendulum hanging on a 3.0 long string, which makes the pendulum swing upward in an arc.
Determine the vertical and horizontal components of the pendulum's displacement.
To determine the vertical and horizontal components of the pendulum's displacement, we'll use the concept of conservation of momentum.
Conservation of momentum states that the initial momentum before the bullet hits the pendulum is equal to the final momentum after the bullet is buried in the pendulum.
Given:
Mass of the bullet (m1) = 25 g = 0.025 kg
Initial velocity of the bullet (v1) = unknown
Mass of the pendulum (m2) = unknown
Final velocity of the pendulum (v2) = 0 (as it comes to rest after the bullet is buried)
Since we know the velocity and mass of the bullet, we can calculate its initial momentum (P1) using the formula:
P1 = m1 * v1
Now, let's calculate the mass of the pendulum using the information given.
Using the formula for pendulum's period:
T = 2π * √(L / g)
where L is the length of the pendulum (3.0 m) and g is the acceleration due to gravity (9.8 m/s^2)
Since the period is given as 3.8 seconds,
3.8 = 2π * √(3 / 9.8)
Solving this equation, we find:
L = 1.142 m
Now, let's calculate the mass of the pendulum using the formula for the period of a simple pendulum:
T = 2π * √(L / g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
Rearranging the formula, we get:
m2 = (T / (2π))^2 * (g / L)
Substituting the values:
m2 = (3.8 / (2π))^2 * (9.8 / 1.142)
Now that we know the mass of the pendulum, we can write the equation for conservation of momentum:
P1 = P2
(m1 * v1) = (m2 * v2)
Since v2 is zero, we can rewrite the equation as:
m1 * v1 = m2 * 0
Simplifying further, we get:
v1 = 0 / m1
Therefore, the vertical component of the pendulum's displacement is zero because there is no change in vertical momentum.
To determine the horizontal component, we need to use the formula for the displacement of a pendulum:
Displacement = L * sin(θ)
where L is the length of the pendulum and θ is the angle through which the pendulum swings.
Since the pendulum swings upward in an arc, θ is positive.
Now, let's find the angle through which the pendulum swings.
Using the formula:
θ = arcsin(v1 / (L * g))
Substituting the values we know:
θ = arcsin(0 / (3.0 * 9.8))
Since the vertical component is zero, θ will also be zero.
Therefore, the horizontal component of the pendulum's displacement is zero as well.
To determine the vertical and horizontal components of the pendulum's displacement, you can use the principles of conservation of momentum and conservation of energy.
Step 1: Calculate the momentum of the bullet before impact.
The momentum of an object is given by the equation:
Momentum = Mass x Velocity
Given:
Mass of the bullet (m) = 25 g = 0.025 kg
Velocity of the bullet (v) = 280 m/s
Momentum of the bullet before impact = m x v
Step 2: Calculate the energy transferred to the pendulum.
The energy transferred to the pendulum is equal to the kinetic energy of the bullet before impact. The kinetic energy is given by the equation:
Kinetic Energy = 1/2 x Mass x Velocity^2
Given:
Mass of the bullet (m) = 0.025 kg
Velocity of the bullet (v) = 280 m/s
Kinetic Energy transferred = 1/2 x m x v^2
Step 3: Determine the vertical height reached by the pendulum.
The energy transferred to the pendulum is converted into potential energy, causing the pendulum to swing upward. The potential energy of an object is given by the equation:
Potential Energy = Mass x Gravity x Height
Given:
Mass of the pendulum (M) = Unknown (to be determined)
Gravity (g) = 9.8 m/s^2
Height reached by the pendulum = Unknown (to be determined)
Potential Energy gained by the pendulum = M x g x height
Step 4: Use conservation of momentum to determine the velocity of the pendulum.
Since momentum is conserved, the momentum of the bullet before impact must be equal to the momentum of the pendulum after impact. The momentum of an object is given by the equation:
Momentum = Mass x Velocity
Given:
Mass of the pendulum (M) = Unknown (to be determined)
Velocity of the pendulum (v') = Unknown (to be determined)
Momentum of the pendulum after impact = M x v'
Step 5: Use conservation of energy to determine the horizontal component of the pendulum's displacement.
The energy transferred to the pendulum is also converted into kinetic energy. The kinetic energy of an object is given by the equation:
Kinetic Energy = 1/2 x Mass x Velocity^2
Given:
Mass of the pendulum (M) = Unknown (to be determined)
Velocity of the pendulum (v') = Unknown (to be determined)
Kinetic Energy gained by the pendulum = 1/2 x M x v'^2
Step 6: Solve the equations simultaneously to determine the unknown variables.
Equating the momentum and energy equations, we have:
M x v' = m x v (conservation of momentum)
1/2 x M x v'^2 = 1/2 x m x v^2 (conservation of energy)
By solving these equations simultaneously, you can find the values of M, v', and height.
This process involves algebraic manipulation and solving equations. By substituting the known values into the equations and solving them, you can determine the vertical and horizontal components of the pendulum's displacement.
Using conservation of momentum, find the velocity of the block/bullet just after impact.
With that velocity, the block/bullet has an initial KEnergy, that will equal the PE at the highest point.
Sketch it out to figure the height the pendulum.