A ballistic pendulum consists of an arm of mass M and length L = 0.485 m. One end of the arm is pivoted so that the arm rotates freely in a vertical plane. Initially, the arm is motionless and hangs vertically from the pivot point. A projectile of the same mass M hits the lower end of the arm with a horizontal velocity of V = 2.17 m/s. The projectile remains stuck to the free end of the arm during their subsequent motion. Find the maximum angle to which the arm and attached mass will swing in each case:
a) The arm is treated as an ideal pendulum, with all of its mass concentrated as a point mass at the free end.
To find the maximum angle to which the arm and attached mass will swing, we can use the principle of conservation of mechanical energy.
Using the principle of conservation of mechanical energy, we can equate the initial mechanical energy of the system to the maximum potential energy at the highest point of the swing.
The initial mechanical energy of the system is given by the sum of the kinetic energy and potential energy before the collision:
Initial mechanical energy = 0.5 * M * V^2 + M * g * L
Where:
M is the mass of the arm and projectile
V is the horizontal velocity of the projectile
g is the acceleration due to gravity
L is the length of the arm
At the highest point of the swing, the system reaches its maximum potential energy. At this point, all the initial kinetic energy has been converted to potential energy.
Maximum potential energy = M * g * L * (1 - cos(theta))
Where:
theta is the maximum angle to which the arm and attached mass swing
Setting the initial mechanical energy equal to the maximum potential energy, we have:
0.5 * M * V^2 + M * g * L = M * g * L * (1 - cos(theta))
Simplifying the equation, we get:
0.5 * V^2 = g * L * (1 - cos(theta))
Rearranging the equation, we can solve for cos(theta):
cos(theta) = 1 - (0.5 * V^2) / (g * L)
Now, we can solve for theta by taking the inverse cosine of both sides:
theta = acos(1 - (0.5 * V^2) / (g * L))
Plugging in the given values, we have:
theta = acos(1 - (0.5 * (2.17)^2) / (9.8 * 0.485))
theta ≈ acos(0.7876)
theta ≈ 39.7 degrees (rounded to one decimal place)
Therefore, the maximum angle to which the arm and attached mass will swing, when treated as an ideal pendulum, is approximately 39.7 degrees.
To find the maximum angle to which the arm and attached mass will swing when the arm is treated as an ideal pendulum, we can apply the principle of conservation of mechanical energy.
The initial energy of the system is purely due to the kinetic energy of the projectile, given by K1 = 0.5 * M * V^2, as the arm is initially motionless.
At the maximum angle, the projectile and the arm will have reached their maximum height, and all their initial kinetic energy will have been converted into gravitational potential energy. Let's call this maximum potential energy PE_max.
Since the arm is treated as an ideal pendulum with all its mass concentrated as a point mass at the free end, its potential energy PE_arm = M * g * L * (1 - cosθ), where g is the acceleration due to gravity and θ is the angle through which the arm swings.
Therefore, at the maximum angle, the total potential energy of the system can be expressed as PE_total = PE_max + PE_arm.
The equation for conservation of mechanical energy can then be written as:
K1 = PE_total
0.5 * M * V^2 = PE_max + M * g * L * (1 - cosθ)
Now, we can solve for the maximum angle θ.
Rearrange the equation:
PE_max = 0.5 * M * V^2 - M * g * L * (1 - cosθ)
Set this expression for PE_max equal to the potential energy of the arm:
0.5 * M * V^2 - M * g * L * (1 - cosθ) = M * g * L * (1 - cosθ)
Cancel out the common term M * g * L * (1 - cosθ) from both sides:
0.5 * V^2 = g * L * (1 - cosθ)
Now, solve for cosθ:
cosθ = 1 - (0.5 * V^2) / (g * L)
Once you have the value of cosθ, you can find the maximum angle θ by taking the inverse cosine:
θ = cos^(-1)(1 - (0.5 * V^2) / (g * L))
Substitute the given values of V, L, and g to find the maximum angle to which the arm and attached mass will swing in this case.