A 0.15kg bullet is fired into a 1.85kg block that is hanging from a 45cm string. What minimum initial bullet speed is required if the bullet and block are to make one complete revolution?
Please show all work. I need this question answered as soon as possible!
at the top of the circle:
v^2/r = g
v^2 = g r = 9.81 * .45 = 4.4145
so at top v = 2.1 m/s
so Kinetic energy at the bottom must be enough to get up there at that speed
(1/2) m v^2 = m g h + (1/2) m (4.4145)
(1/2) v^2 = 9.81(0.90) + 2.207
v = 4.7 m/s at bottom after crash
NOW do your usual conservation of momentum thing at the bottom with the bullet in the block
what would I solve for when I do the conservation of momentum at the bottom of the block with the bullet?
LOL, figured you knew how to do all that
initial bullet speed = V
that is what we want
we know final speed is 4.7
conservation of momentum:
.15 V = (.15+1.85)(4.7)
the change in energy from the bottom of the circle to the top is... (m g h)
h = 90 cm
the centripetal force at the top is only gravitational... m g
velocity at top ... m v^2 / r = m g
___ v = √(r g) ... r is 45 cm
the KE at the bottom equals the KE
___ at the top, PLUS (m g h)
use the KE at the bottom to find the
___ block/bullet bottom velocity
momentum is conserved in the block/bullet collision
V bullet * M bullet equals
V block/bullet * M block/bullet
Thank you so much!!!
You are welcome.
To find the minimum initial bullet speed required for the bullet and block to make one complete revolution, we can solve it using the principles of conservation of angular momentum.
1. First, we need to calculate the moment of inertia of the system. The moment of inertia of an object rotating around a fixed axis is given by the formula: I = m * r^2, where m is the mass of the object and r is the distance from the axis of rotation.
The moment of inertia of the bullet can be approximated to a point mass: I_bullet = m_bullet * r_bullet^2 = 0.15 kg * (0.45 m)^2 = 0.0091125 kg * m^2.
The moment of inertia of the block hanging from the string can be approximated as a point mass at the center of the block: I_block = m_block * r_block^2 = 1.85 kg * (0.225 m)^2 = 0.09345625 kg * m^2.
The total moment of inertia of the system is the sum of the bullet and block's moment of inertia: I_total = I_bullet + I_block = 0.0091125 kg * m^2 + 0.09345625 kg * m^2 = 0.10256875 kg * m^2.
2. Next, we need to determine the angular momentum of the system in terms of the bullet's initial speed, v_bullet. Angular momentum is the product of the moment of inertia and the angular velocity. In this case, the angular velocity is equal to the number of revolutions (1) multiplied by 2π radians (a full circle).
Angular momentum of the system = I_total * ω = I_total * (2π / T), where T is the time it takes to complete one revolution.
Since the bullet and block are attached to the string and the string length remains constant, we can relate the angular velocity to the linear velocity using the formula: v_bullet = ω * r_bullet, where r_bullet is the radius of the circle.
Substituting the value of angular velocity (ω) and linear velocity (v_bullet) into the angular momentum equation, we have:
I_total * (2π / T) = I_total * v_bullet / r_bullet
3. Finally, we can solve for v_bullet. Rearranging the equation, we get:
v_bullet = (2π * r_bullet) / T
To find the minimum initial bullet speed, we want T to be the minimum time it takes for a complete revolution. This occurs when the period (T) is equal to the time it takes for the bullet to travel the string's length (2r_bullet), divided by its speed:
T = (2 * r_bullet) / v_bullet
Substituting this value of T back into the equation, we have:
v_bullet = (2π * r_bullet) / ((2 * r_bullet) / v_bullet)
Simplifying the equation, we get:
v_bullet^2 = (2π * r_bullet)^2 / (2 * r_bullet)
v_bullet^2 = (4π^2 * r_bullet^2) / (2 * r_bullet)
v_bullet^2 = 2π^2 * r_bullet
Taking the square root of both sides, we find:
v_bullet = √(2π^2 * r_bullet)
v_bullet = √(2 * 3.14^2 * 0.45 m)
Calculating the values, we get:
v_bullet ≈ √(2 * 3.14^2 * 0.45 m) ≈ 2.76 m/s
Therefore, the minimum initial bullet speed required for the bullet and block to make one complete revolution is approximately 2.76 m/s.
Please note that this is a simplified calculation assuming no air resistance and friction in the system. In practice, additional factors may affect the actual speed required.