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)?
at what height is an object that weighs 490 N gravitational potential energy is 4900 N-m/
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To determine how long it takes for the door to fully open, we need to consider the conservation of angular momentum.
Angular momentum is the product of moment of inertia (I) and angular velocity (ω). In this case, the moment of inertia is determined by the mass (m) and width (w) of the door.
The moment of inertia (I) for a thin rod rotating about its center is given by the formula:
I = (m * w^2) / 12
Given:
Mass (m) of the door = 15 kg
Width (w) of the door = 1 m
Substituting the given values into the formula, we can calculate the moment of inertia (I):
I = (15 kg * (1 m)^2) / 12 = 1.25 kg*m^2
The collision involving the ball causes a change in angular momentum, which leads to the door rotating. The ball has a momentum (p) given by the product of its mass (m ball) and velocity (v ball):
p = m ball * v ball
Given:
Mass (m ball) of the ball = 400 g = 0.4 kg
Velocity (v ball) of the ball = 35 m/s
Therefore, the momentum (p) of the ball is:
p = 0.4 kg * 35 m/s = 14 kg*m/s
The change in angular momentum (ΔL) of the door is equal to the impulse provided by the ball. Since the collision is elastic, the momentum of the ball before and after the collision is the same. Hence, the change in angular momentum is equal to 2 times the momentum of the ball.
ΔL = 2 * p
Substituting the calculated value for p, we find:
ΔL = 2 * 14 kg*m/s = 28 kg*m/s
Now, we can determine the angular velocity (ω) with which the door rotates. We can use the equation:
ΔL = I * ω
Rearranging the equation gives us:
ω = ΔL / I
Substituting the calculated values, we find the angular velocity (ω):
ω = 28 kg*m/s / 1.25 kg*m^2 = 22.4 rad/s
To calculate the time it takes for the door to fully open, we can use the equation:
θ = ω * t
Where:
θ is the angle through which the door rotates (90 degrees or π/2 radians),
ω is the angular velocity, and
t is the time it takes for the door to rotate.
Rearranging the equation gives us:
t = θ / ω
Substituting the value for θ, we have:
t = π/2 radians / 22.4 rad/s
Using a calculator, we find:
t = 0.141 seconds
Therefore, it takes approximately 0.141 seconds for the door to fully open (rotate 90 degrees).