A 84-kg fireman slides down a 4-m pole by applying a frictional force of 416 N against the pole with his hands. If he slides from rest, how fast is he moving once he reaches the ground?
To find the speed at which the fireman is moving once he reaches the ground, we can use the principle of conservation of mechanical energy. The mechanical energy of the system is conserved, assuming there is no dissipative force, such as air resistance.
First, we need to calculate the gravitational potential energy at the top of the pole. Using the formula:
Potential Energy = mass * gravity * height
where the mass of the fireman is 84 kg, the acceleration due to gravity is approximately 9.8 m/s², and the height of the pole is 4 m:
Potential Energy = 84 kg * 9.8 m/s² * 4 m
= 3292.8 J
Since there is no initial kinetic energy (starting from rest), the total mechanical energy of the system is equal to the initial potential energy at the top of the pole.
Next, we need to calculate the work done by the frictional force. The work done is given by the formula:
Work = force * distance * cos(θ)
where the force is 416 N, the distance is 4 m (since the fireman slides down the entire length of the pole), and θ is the angle between the force and the direction of motion (which is 0 degrees because the force and motion are in the same direction):
Work = 416 N * 4 m * cos(0°)
= 1664 J
The work done by the frictional force is negative since it opposes the motion. Therefore, the change in mechanical energy is equal to the negative work done by the frictional force:
Change in Mechanical Energy = -1664 J
Finally, we can calculate the final kinetic energy at the bottom of the pole using the equation:
Final Kinetic Energy = Initial Potential Energy + Change in Mechanical Energy
Initial Potential Energy = 3292.8 J
Final Kinetic Energy = 3292.8 J - 1664 J
= 1638.8 J
The final kinetic energy is the same as the initial potential energy because mechanical energy is conserved. To find the final velocity (speed), we can use the equation:
Final Kinetic Energy = (1/2) * mass * velocity²
Rearranging the equation:
velocity² = (2 * Final Kinetic Energy) / mass
velocity² = (2 * 1638.8 J) / 84 kg
velocity² = 39.02 m²/s²
Taking the square root of both sides:
velocity = √(39.02 m²/s²)
velocity ≈ 6.25 m/s
Therefore, the fireman is moving at approximately 6.25 m/s once he reaches the ground.