A person pushes a 16.0 kg lawn mower at constant speed with a force of 70.0 N directed along the handle, which is at an angle of è = 45.0° to the horizontal. Calculate the force the person must exert on the lawn mower to accelerate it from rest to 1.2 m/s in 2.0 seconds (assuming the same retarding force which is 70cos45).
We can start by finding the horizontal component of the force that the person exerts on the lawn mower using the given angle. We can call this horizontal force F_x.
F_x = F * cos(45)
F_x = 70 * cos(45)
F_x ≈ 49.5 N
We know that the retarding force acting on the lawn mower is 70*cos(45), which is equal to the horizontal component of the applied force (F_x) when the lawn mower is moving at a constant speed. So, the net force acting on the lawn mower when it's accelerating is the same as when it's moving at a constant speed, which is about 49.5 N.
Now we can use Newton's second law of motion, F_net = m * a, to find the acceleration (a) of the lawn mower.
49.5 = 16 * a
a ≈ 3.1 m/s^2
Now that we have the acceleration, we can use the following kinematic equation to find the final velocity (v) of the lawn mower after 2 seconds of acceleration, starting from rest (initial velocity, u = 0):
v = u + a*t
v = 0 + (3.1 * 2)
v ≈ 6.2 m/s
However, the problem states that the final velocity of the lawn mower should be 1.2 m/s. So, we need to adjust the net force to achieve this final velocity.
We can use the same kinematic equation with the desired final velocity, v = 1.2 m/s:
1.2 = 0 + a*2
a ≈ 0.6 m/s^2
Now we can use Newton's second law to find the required net force:
F_net = m * a
F_net = 16 * 0.6
F_net ≈ 9.6 N
Since the required net force is less than the initial net force (49.5 N), we need to find the difference in force that the person must exert to achieve the desired final velocity.
∆F = 49.5 - 9.6
∆F ≈ 39.9 N
Therefore, the person must exert an additional force of about 39.9 N on the lawn mower to accelerate it from rest to 1.2 m/s in 2 seconds.
To calculate the force required to accelerate the lawn mower, we'll need to apply Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
Given:
Mass of the lawn mower (m) = 16.0 kg
Initial speed (v_i) = 0 m/s
Final speed (v_f) = 1.2 m/s
Time taken (t) = 2.0 s
The retarding force acting on the lawn mower is given as 70 * cos(45) N.
Step 1: Calculate the acceleration (a):
We can use the formula for acceleration, which is the change in velocity divided by the time taken: a = (v_f - v_i) / t.
Plugging in the values: a = (1.2 m/s - 0 m/s) / 2.0 s = 0.6 m/s^2.
Step 2: Calculate the net force (F_net):
The net force acting on the lawn mower is the force required to overcome the retarding force. So, F_net = retarding force.
F_net = 70 N * cos(45°) = 70 N * 0.707 ≈ 49.49 N.
Step 3: Calculate the force required to accelerate the lawn mower (F):
We can use Newton's second law of motion: F = m * a.
Plugging in the values: F = 16.0 kg * 0.6 m/s^2 = 9.6 N.
Therefore, the person must exert a force of approximately 9.6 N on the lawn mower to accelerate it from rest to 1.2 m/s in 2.0 seconds (assuming the same retarding force of 70 * cos(45)).
To solve this problem, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. We can calculate the net force acting on the lawn mower using the given information.
1. Calculate the net force required to accelerate the lawn mower:
First, we need to find the acceleration of the lawn mower using the formula:
acceleration = change in velocity / time
Given:
Initial velocity (u) = 0 m/s (since the lawn mower is at rest)
Final velocity (v) = 1.2 m/s
Time (t) = 2.0 seconds
acceleration = (1.2 - 0) / 2.0
acceleration = 0.6 m/s^2
Next, we can calculate the net force using Newton's second law:
net force = mass × acceleration
Given:
mass (m) = 16.0 kg
acceleration = 0.6 m/s^2
net force = 16.0 kg × 0.6 m/s^2
net force = 9.6 N
Therefore, the net force required to accelerate the lawn mower is 9.6 N.
2. Calculate the force the person must exert on the lawn mower:
To calculate the force the person must exert, we need to account for the retarding force, which is given as 70cos45.
retarding force = 70 N × cos(45°)
Using the value of cos(45°) = 0.7071, we have:
retarding force = 70 N × 0.7071
retarding force = 49.4967 N
The force the person must exert on the lawn mower will be equal to the sum of the net force required to accelerate the lawn mower and the retarding force:
force exerted = net force + retarding force
force exerted = 9.6 N + 49.4967 N
force exerted ≈ 59.1 N
Therefore, the person must exert a force of approximately 59.1 N on the lawn mower to accelerate it from rest to 1.2 m/s in 2.0 seconds.