A 15 kg, 1m wide door which has frictionless hinges is closed but unlocked. A 400 g ball hits the exact middle of the door at a velocity of 35 m/s and bounces off elastically, thereby causing the door to slowly swing open. How long in seconds does it take for the door to fully open (rotate 90 degrees)?
1.68?
how did you get 1.68?
Since it is an elastic impact, we can equate energies.
To find the time it takes for the door to fully open, we need to calculate the angular acceleration of the door first. Here are the steps to solve the problem:
1. Calculate the moment of inertia of the door:
The moment of inertia (I) depends on the shape and mass distribution of the object. For a rectangular door rotating about its center, the moment of inertia can be calculated using the formula: I = (1/3) * M * L^2, where M is the mass and L is the length of the door.
In this case, the mass (M) of the door is 15 kg, and the length (L) is given as 1 m. Plugging these values into the formula, we have:
I = (1/3) * 15 kg * (1 m)^2 = 5 kg·m²
2. Calculate the initial angular momentum of the ball:
Angular momentum (L) is the product of rotational inertia (I) and angular velocity (ω): L = I * ω. Initially, the door is at rest, so the initial angular velocity is zero.
Therefore, the initial angular momentum (L_initial) of the ball can be calculated as:
L_initial = I * ω_initial = 5 kg·m² * 0 = 0 kg·m²/s
3. Calculate the final angular momentum of the ball:
After the ball hits the door, it exerts a force causing the door to rotate. Due to the conservation of angular momentum, the final angular momentum (L_final) of the ball will be equal in magnitude but in the opposite direction to the initial angular momentum.
Since the ball is initially at rest (ω_final = 0), the final angular momentum can be calculated as:
L_final = -L_initial = -0 kg·m²/s = 0 kg·m²/s
4. Calculate the change in angular momentum:
The change in angular momentum (ΔL) can be calculated as:
ΔL = L_final - L_initial = 0 kg·m²/s - 0 kg·m²/s = 0 kg·m²/s
5. Calculate the impulse:
Impulse (J) is the change in momentum and is equal to the change in angular momentum (ΔL): J = ΔL
6. Calculate the torque applied by the impulse:
The torque (τ) acting on the door is equal to the impulse applied divided by the time of collision (Δt). Since the ball bounces off elastically, Δt can be calculated using the following formula: Δt = (2 * velocity) / distance.
Given that the velocity of the ball is 35 m/s and the distance is 1 m (door width), we can calculate Δt:
Δt = (2 * 35 m/s) / 1 m = 70 s
Now we can calculate the torque:
τ = J / Δt = ΔL / Δt = 0 kg·m²/s / 70 s = 0 N·m
7. Calculate the angular acceleration:
The angular acceleration (α) of the door can be calculated using the formula: τ = I * α, where τ is the torque and I is the moment of inertia.
Plugging in the values, we have:
0 N·m = 5 kg·m² * α
α = 0 N·m / 5 kg·m² = 0 rad/s²
Since the torque is zero, the angular acceleration is also zero. This means the door doesn't rotate under the force of the ball.
Therefore, the door will not fully open, and the time it takes to rotate 90 degrees is infinite or never.