A 3.00-kg block starts from rest at the top of a 31.0° incline and slides 2.00 m down the incline in 1.40 s.
(a) Find the acceleration of the block.
m/s2
(b) Find the coefficient of kinetic friction between the block and the incline.
(c) Find the frictional force acting on the block.
N
(d) Find the speed of the block after it has slid 2.00 m.
m/s
a) x = 1/2a t^2
c) mg sin31 - Ff = ma
b) Fn = mgcos31, mu = Ff/Fn
d) vf = at
To solve these types of problems, we can use the principles of Newton's laws of motion and kinematics. Let's break down each part of the problem step by step:
(a) Find the acceleration of the block:
1. Draw a free-body diagram showing all the forces acting on the block. In this case, the forces are the gravitational force (mg) downward, the normal force (N) perpendicular to the incline, and the frictional force (f) opposing the motion.
2. Decompose the gravitational force into its components. The force component parallel to the incline is m * g * sin(θ), where θ is the angle of the incline.
3. Apply Newton's second law in the direction parallel to the incline: ΣF = m * a, where ΣF is the sum of all the forces acting on the block parallel to the incline.
ΣF = m * a
m * g * sin(θ) - f = m * a
4. Rearrange the equation and solve for the acceleration (a):
a = (m * g * sin(θ) - f) / m
(b) Find the coefficient of kinetic friction between the block and the incline:
1. We can use the equation for kinetic friction: f = μ * N, where μ is the coefficient of kinetic friction and N is the normal force.
2. The normal force can be calculated as N = m * g * cos(θ), where θ is the angle of the incline.
3. Substitute the expression for the normal force into the equation for kinetic friction:
f = μ * (m * g * cos(θ))
4. Rearrange the equation and solve for the coefficient of kinetic friction (μ):
μ = f / (m * g * cos(θ))
(c) Find the frictional force acting on the block:
1. Substitute the expression for the normal force into the equation for kinetic friction:
f = μ * (m * g * cos(θ))
2. Using the coefficient of kinetic friction (μ) obtained in the previous step, calculate the frictional force (f).
(d) Find the speed of the block after it has slid 2.00 m:
1. We can use the kinematic equation: v² = u² + 2 * a * s, where v is the final velocity, u is the initial velocity (which is zero in this case), a is the acceleration, and s is the displacement.
2. Rearrange the equation and solve for the final velocity (v):
v = √(u² + 2 * a * s)
Since the initial velocity is zero, the equation simplifies to:
v = √(2 * a * s)
Substitute the values of the acceleration (a) and displacement (s) to obtain the final velocity (v).
Now let's work through each part of the problem to find the answers.