at a playground a child slides that makes a 42 degrees angle with the horizontal direction.The coefficient of kinetic friction for this motion is 0.20 What is the magnitude of her acceleration during her sliding
To find the magnitude of the acceleration, we can first calculate the net force acting on the child as she slides down the slide. The net force can be expressed as the product of the mass (m) and acceleration (a): F_net = m * a.
Next, we need to break down the forces acting on the child. There are two main forces to consider: the force of gravity (F_gravity) and the force of friction (F_friction). The force of gravity can be calculated using the equation:
F_gravity = m * g
Where:
m = mass of the child
g = acceleration due to gravity (approximately 9.8 m/s^2).
The force of friction can be calculated using the equation:
F_friction = μ * N
Where:
μ = coefficient of kinetic friction
N = normal force
In this case, the normal force is equal to the component of the weight perpendicular to the slide, which can be calculated as:
N = m * g * cos(42)
Therefore, the force of friction will be:
F_friction = 0.20 * m * g * cos(42)
Since the child is sliding down the slide, the force of friction acts opposite to the direction of motion. Therefore, the net force can be calculated as:
F_net = F_gravity - F_friction
Now, we equate the net force to the mass times the acceleration:
m * a = F_net
Substituting the expressions for F_gravity and F_friction:
m * a = (m * g) - (0.20 * m * g * cos(42))
Simplifying the equation:
a = g - (0.20 * g * cos(42))
Now, we can substitute the known values into the equation:
a = 9.8 m/s^2 - (0.20 * 9.8 m/s^2 * cos(42))
Calculating the numerical value:
a ≈ 9.8 m/s^2 - (0.20 * 9.8 m/s^2 * 0.7431)
a ≈ 9.8 m/s^2 - 1.448 m/s^2
a ≈ 8.352 m/s^2
Therefore, the magnitude of the child's acceleration while sliding down the slide is approximately 8.352 m/s^2.
To determine the magnitude of the child's acceleration while sliding down the playground, you can use the following steps:
Step 1: Identify the forces acting on the child. In this case, there are two main forces: gravity acting downward (mg) and the kinetic friction opposing the motion.
Step 2: Find the force of gravity acting on the child. The force of gravity is given by the formula Fg = mg, where m is the mass of the child and g is the acceleration due to gravity (approximately 9.8 m/s²).
Step 3: Determine the angle of the slide (42 degrees) and decompose the force of gravity into its components. The force of gravity can be split into two components: the force acting parallel to the slide (mg sin θ) and the force acting perpendicular to the slide (mg cos θ).
Step 4: Calculate the force of kinetic friction (Fk) using the formula Fk = μk * N, where μk is the coefficient of kinetic friction and N is the normal force. The normal force (N) is equal to the perpendicular component of the force of gravity (mg cos θ).
Step 5: Now that you have the force of kinetic friction, you can calculate the net force acting on the child. The net force is given by the formula Fnet = Fg sin θ - Fk.
Step 6: Finally, calculate the acceleration (a) using Newton's second law, Fnet = ma, where m is the mass of the child. Rearrange the equation to solve for acceleration: a = Fnet / m.
By following these steps, you can find the magnitude of the child's acceleration while sliding down the playground.