The man pushes and pulls with a force of 200 N. The child and sled
combo have a mass of 30 kg and the coefficient of kinetic friction is
0.15.The angle between the slide and the ground is 30 degrees. For each case:
What is the frictional force opposing his efforts?
What is the acceleration of the child?
To solve this problem, we need to use Newton's second law:
ΣF = ma
where ΣF is the net force acting on the child and sled, m is the mass of the child and sled combo, and a is the acceleration of the child and sled combo.
First, we need to find the frictional force opposing the man's efforts. This can be found using the formula:
f_friction = μ_k * N
where μ_k is the coefficient of kinetic friction, and N is the normal force acting on the child and sled combo. The normal force is equal to the weight of the child and sled combo, which can be found using the formula:
N = mg
where g is the acceleration due to gravity (9.8 m/s^2).
N = (30 kg)(9.8 m/s^2) = 294 N
f_friction = (0.15)(294 N) = 44.1 N
Therefore, the frictional force opposing the man's efforts is 44.1 N.
Next, we need to find the acceleration of the child and sled combo. To do this, we need to find the net force acting on the child and sled combo. The forces acting on the child and sled combo are:
- The force exerted by the man, which is 200 N.
- The frictional force opposing motion, which is 44.1 N.
- The weight of the child and sled combo, which is 294 N (downward).
The weight of the child and sled combo can be resolved into two components: one parallel to the slide and one perpendicular to the slide. The component parallel to the slide is:
F_g_parallel = mg * sinθ
where θ is the angle between the slide and the ground.
F_g_parallel = (30 kg)(9.8 m/s^2) * sin(30°) = 147 N
Therefore, the net force acting on the child and sled combo is:
ΣF = F_applied - f_friction - F_g_parallel
ΣF = (200 N) - (44.1 N) - (147 N) = 8.9 N (to the right)
Finally, we can find the acceleration of the child and sled combo using Newton's second law:
ΣF = ma
8.9 N = (30 kg) a
a = 0.3 m/s^2
Therefore, the acceleration of the child and sled combo is 0.3 m/s^2.