A small boy is trying to open a 50 kg door with a height of 2 meters and a width of 75 centimeters. If he applies a force of 5 Newtons at the door handle (5 cm from the edge), how long will it take to open the door so that it's completely open (perpindicular to when it was closed)?
torque= I * acceleration.
torque= .75*5
I= 1/3*50*.75^5
solve for acceleration.
To calculate the time it takes to open the door, we need to determine the torque applied to the door by the boy's force and then use the torque to find the angular acceleration of the door. Finally, we can calculate the time using the equation of motion for rotational motion.
Step 1: Calculate the torque:
Torque (τ) is given by the formula τ = r * F, where r is the distance between the point where the force is applied and the axis of rotation, and F is the applied force.
In this case, the force applied is 5 Newtons, and the distance from the force to the axis of rotation (door hinge) is 5 cm or 0.05 meters.
So, the torque applied to the door is τ = 0.05 meters * 5 Newtons = 0.25 Nm.
Step 2: Calculate the moment of inertia:
The moment of inertia (I) depends on the mass distribution of the door. Assuming the door is a uniform rectangular plate rotating about one edge, the moment of inertia can be calculated as:
I = (1/3) * m * L^2, where m is the mass of the door and L is the length of the door.
The mass of the door is given as 50 kg, and the length (height) of the door is 2 meters.
So, the moment of inertia of the door is I = (1/3) * 50 kg * (2 meters)^2 = 66.67 kg*m^2.
Step 3: Calculate the angular acceleration:
The torque applied to a rotating object is given by the equation τ = I * α, where α is the angular acceleration.
Rearranging the equation, α = τ / I.
Substituting the values, α = 0.25 Nm / 66.67 kg*m^2 = 0.00375 rad/s^2.
Step 4: Calculate the time to open the door:
The relationship between angular acceleration (α), initial angular velocity (ω0), final angular velocity (ω), and time (t) in rotational motion is given by the equation:
ω = ω0 + α * t.
Initially, the door is at rest, so the initial angular velocity (ω0) is 0.
To open the door completely, the final angular velocity (ω) should be 90 degrees per second, as the door needs to rotate 90 degrees to be perpendicular to the closed position.
Converting 90 degrees to radians, ω = (90 degrees) * (π radians / 180 degrees) = π / 2 rad/s.
Substituting the values, 0 = 0 + 0.00375 rad/s^2 * t.
Simplifying, t = 0 / 0.00375 rad/s^2 = 0 seconds.
Thus, it will take the door 0 seconds to open completely.
Please note that in this idealized scenario, we are assuming no friction or other external forces affecting the behavior of the door. In practice, various factors like friction, door weight distribution, and the strength of the door hinge may affect the actual time taken to open the door.