A person opens a door by applying a 11-N force perpendicular to it at a distance 0.90m from the hinges. The door is pushed wide open (to 120 ∘ ) in 2.2s How much work was done? What was the average power delivered?
work=force*distance
=11N*.9*2PI/3
power=above/time
120/360 = 1/3 of circumference
so distance pushed = d = (1/3) 2 pi (.9)
work = force * d = 11 *d Joules
power = work/time = 11 * d / 2.2 in Watts
To find the work done, we can first calculate the torque applied to the door, and then use it to determine the angular displacement of the door during the opening.
1. Calculate the torque:
The torque (τ) applied to the door can be calculated using the formula:
τ = force × distance × sin(θ),
where
force = 11 N (given),
distance = 0.90 m (given),
θ = 90° (since the force is perpendicular to the door).
Plugging in the values, we get:
τ = 11 N × 0.90 m × sin(90°).
Since sin(90°) equals 1, the torque simplifies to:
τ = 11 N × 0.90 m.
τ = 9.9 N⋅m.
2. Calculate the angular displacement:
The angular displacement (θ) of the door can be determined using the formula:
θ = ω × t,
where
ω = angular velocity,
t = time taken to open the door.
Since the door is pushed wide open to 120° (given), the angular displacement is 120°.
Plugging in the values, we get:
120° = ω × 2.2 s.
3. Calculate the average angular velocity:
To find the average angular velocity (ω), we rearrange the formula as follows:
ω = θ / t.
Plugging in the values, we get:
ω = 120° / 2.2 s.
Converting degrees to radians, we have:
ω = (120° × π) / (180° × 2.2 s).
ω = (π / 3) rad/s.
4. Calculate the work done:
The work done (W) can be calculated using the formula:
W = τ × θ,
where
τ = torque (9.9 N⋅m),
θ = angular displacement (120° or π/3 rad).
Plugging in the values, we get:
W = 9.9 N⋅m × (π / 3) rad.
W ≈ 10.33 J. (rounded to two decimal places)
Therefore, the work done is approximately 10.33 J.
5. Calculate the average power delivered:
The average power (P) can be calculated using the formula:
P = W / t,
where
W = work done (10.33 J),
t = time taken to open the door (2.2 s).
Plugging in the values, we get:
P = 10.33 J / 2.2 s.
P ≈ 4.70 W. (rounded to two decimal places)
Therefore, the average power delivered is approximately 4.70 W.
To determine the work done and average power delivered, we can use the following formulas:
1. Work (W) = Force (F) * Distance (d) * cos(θ)
2. Average Power (P_avg) = Work / Time
Where:
- Force (F) = 11 N (given)
- Distance (d) = 0.90 m (given)
- θ (theta) = 120° (given)
- Time (t) = 2.2 s (given)
Now, let's calculate the work done:
1. Calculate the horizontal component of the force:
F_horizontal = F * cos(θ)
F_horizontal = 11 N * cos(120°)
F_horizontal ≈ -5.5 N (negative because it is in the opposite direction)
2. Calculate the work done:
W = F_horizontal * d
W = -5.5 N * 0.90 m
W ≈ -4.95 Joules
Therefore, the work done is approximately -4.95 Joules.
Next, let's calculate the average power delivered:
3. Calculate the average power:
P_avg = W / t
P_avg = -4.95 J / 2.2 s
P_avg ≈ -2.25 Watts
Therefore, the average power delivered is approximately -2.25 Watts.
Note: The negative sign indicates that the work done and power delivered are in the opposite direction of the applied force. It represents the energy being taken from the system (door) rather than being added to it.