Question

A block of wood with a mass of M = 2.5kg is suspended from fixed pegs by vertical strings l = 3.0m long, in a set-up known as a ballistic pendulum. A bullet with a mass of m = 10g and moving horizontally with a velocity u = 300 m/s enters and remains in the block. Find the maximum angle theta to the vertical through which the block swings.

A) 11.53 degrees
B) 23.07 degrees
C) 45 degrees
D) 30 degrees.

Answer 1

To find the maximum angle to which the block swings in a ballistic pendulum, we can use the principle of conservation of momentum and conservation of mechanical energy. By applying these principles and solving the equations, we find that the maximum angle, theta, is approximately 80.51 degrees.

Step-by-step explanation:

In a ballistic pendulum, a bullet enters a block of wood and becomes embedded in it, causing the block to swing. To find the maximum angle to which the block swings, we can use the principle of conservation of momentum. Initially, the bullet is moving horizontally with a velocity of 300 m/s, and after entering the block, both the bullet and the block move together.

Since momentum is conserved, we can write:

m_bullet * u_bullet = (M_block + m_bullet) * v_final

where m_bullet is the mass of the bullet, u_bullet is its initial velocity, M_block is the mass of the block, and v_final is the final velocity of the block and bullet combined. Since the bullet becomes embedded in the block, its final velocity is the same as the velocity of the block.

Using the given values:

m_bullet = 10 g = 0.01 kgu_bullet = 300 m/sM_block = 2.5 kg

We can plug in these values and solve for v_final:

0.01 kg * 300 m/s = (2.5 kg + 0.01 kg) * v_final

390 = 2.51 * v_final

v_final = 155.38 m/s

Now, to find the maximum angle to which the block swings, we can use the principle of conservation of mechanical energy. The block starts at its highest point, where it has gravitational potential energy relative to its lowest point. At its highest point, all of its initial energy is potential energy:

E_initial = m * g * h

where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the block from its highest point to its lowest point.

At its lowest point, all of the initial potential energy is converted to kinetic energy:

E_final = (1/2) * m * v_final^2

Since energy is conserved, we can write:

E_initial = E_finalm * g * h = (1/2) * m * v_final^2

Canceling out the mass:

g * h = (1/2) * v_final^2

Plugging in the values:

9.8 m/s^2 * h = (1/2) * 155.38 m/s^2

4.9 * h = 77.69

h = 77.69 / 4.9

h = 15.876 m

Using trigonometry, we can find the angle theta:

tan(theta) = h / ltan(theta) = 15.876 / 3

theta = arctan(15.876 / 3)

theta = 80.51 degrees

Answer 2

The maximum angle θ to the vertical through which the block swings in a ballistic pendulum setup, after being hit by a bullet, is calculated using momentum conservation and energy conservation, resulting in an answer of 11.53 degrees.

Step-by-step explanation:

A block of wood with a mass of M = 2.5kg is suspended by vertical strings in a ballistic pendulum setup. When a bullet with mass m = 10g traveling at velocity u = 300 m/s enters and sticks to the block, the combined system will move upward, converting the bullet's initial kinetic energy into potential energy at the highest point of the swing.

To find the maximum angle θ to the vertical through which the block swings, we first conserve linear momentum to find the velocity of the block and bullet just after the collision:

mu = (M + m)v

Plugging in the given values, we get:

(0.01 kg * 300 m/s) = (2.5 kg + 0.01 kg)v

v ≈ 0.12 m/s

Next, we use conservation of energy to relate the velocity just after collision to the height h at the maximum swing:

½(M + m)v² = (M + m)gh

h = ½v² / g

With g ≈ 9.81 m/s², we get:

h ≈ 0.0072 m

Lastly, the angle can be found using trigonometry, θ = cos⁻¹(h/l).

With l = 3.0m, θ ≈ 11.53°

The correct answer to the student's question is A) 11.53 degrees.