A person pushes a 14.0-kg lawn mower at constant speed with a force of F = 88.0 N directed along the handle, which is at

an angle of 45.0º to the horizontal (Fig. 4–45). (a) Draw the free-body diagram showing all forces acting on the mower.
Calculate (b) the horizontal friction force on the mower, then (c) the normal force exerted vertically upward on the mower
by the ground. (d) What force must the person exert on the lawn mower to accelerate it from rest to 1.5 m s in 2.5 seconds,
assuming the same friction force?

Marisol pushes a 3.0-kg box 7.0m across the floor with a force of 24N. How much work does Marisol do on the box

Well, it seems like the lawn mower is going through some physics workout! Let's see if I can help you with these questions.

(a) The free-body diagram for the lawnmower would have a force of gravity acting downwards, a normal force acting upwards, a friction force opposing the motion, and a force caused by the person pushing along the handle at an angle.

(b) To calculate the horizontal friction force, we need to consider the angle of 45º. Remember trigonometry? Well, the horizontal component of the force will be F * cos(45º). So go ahead and crunch those numbers!

(c) The normal force exerted by the ground will be equal to the vertical component of the force, which would be F * sin(45º). Don't forget your math formulas!

(d) To calculate the force required to accelerate the lawn mower from rest to 1.5 m/s in 2.5 seconds, we can use Newton's second law: F = m * a. The mass of the lawn mower is given as 14.0 kg, and the acceleration can be calculated by dividing the change in velocity by the time. So, put on your problem-solving hat and find that force!

Remember, these answers might bring a smile to your face, but they're not clowning around.

To solve this problem, we will use Newton's second law, which states that the net force on an object is equal to the product of its mass and acceleration, Fnet = m * a. We can also use the equations for calculating friction force and normal force.

(a) First, let's draw a free-body diagram showing all the forces acting on the lawn mower:
- The weight of the lawn mower acts vertically downward and can be calculated as W = m * g, where m is the mass of the mower and g is the acceleration due to gravity.
- The force applied by the person acts at an angle of 45.0º to the horizontal. We can break it into horizontal and vertical components:
- The horizontal component is F * cos(45º).
- The vertical component is F * sin(45º).
- There is a friction force opposing the motion of the mower, denoted by f.
- The normal force is the force exerted by the ground vertically upward on the mower.

(b) In order to find the friction force, we need to know the acceleration of the mower. Since the mower is moving at a constant speed, the acceleration is zero. This means that the net force on the mower is also zero. Therefore, the friction force must be equal in magnitude and opposite in direction to the horizontal component of the force applied by the person. So, the horizontal friction force (f) is equal to F * cos(45º).

(c) The normal force (N) exerted vertically upward by the ground can be calculated using Newton's second law in the vertical direction. Since the mower is not accelerating vertically, the net force in the vertical direction is zero. Therefore, the normal force must balance the vertical component of the applied force plus the weight of the mower. So, N - F * sin(45º) - W = 0.

(d) To find the force the person must exert to accelerate the mower from rest to 1.5 m/s in 2.5 seconds, we can use the equation of motion: F = m * a. Rearranging the equation, we have F = (m * v) / t, where v is the final velocity and t is the time taken. Assuming the friction force remains the same, the force exerted by the person must overcome both the friction force and the force required to accelerate the mower. So, the force is equal to the sum of the horizontal friction force we calculated earlier and the force required to accelerate the mower.

Now let's calculate the values:

(a) The free-body diagram:
- Weight (W) = m * g = 14.0 kg * 9.8 m/s² = 137.2 N
- Horizontal force component = F * cos(45º) = 88.0 N * cos(45º) = 62.2 N
- Vertical force component = F * sin(45º) = 88.0 N * sin(45º) = 62.2 N

(b) Horizontal friction force:
- f = F * cos(45º) = 88.0 N * cos(45º) ≈ 62.2 N

(c) Normal force:
- N - F * sin(45º) - W = 0
- N = F * sin(45º) + W = 88.0 N * sin(45º) + 137.2 N ≈ 199.5 N

(d) Force required to accelerate the mower:
- F = (m * v) / t = (14.0 kg * 1.5 m/s) / 2.5 s = 8.4 N

So, the force the person must exert on the lawn mower to accelerate it to 1.5 m/s in 2.5 seconds, assuming the same friction force, is approximately 62.2 N + 8.4 N = 70.6 N.

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