A 3.5 kg block, initially in motion, is pushed along a horizontal floor by a force F of magnitude 17 N at an angle θ = 35° with the horizontal (Figure 6-20). The coefficient of kinetic friction between the block and floor is 0.25. (Assume the positive direction is to the right.)
Figure 6-20
(a) Calculate the magnitude of the frictional force on the block from the floor.
N
(b) Calculate the magnitude of the block's acceleration.
m/s2
To calculate the magnitude of the frictional force on the block from the floor, you can use the formula:
frictional force (Ff) = coefficient of friction (μ) * normal force (N)
The normal force (N) is equal to the weight of the block, which can be calculated using the formula:
weight (W) = mass (m) * acceleration due to gravity (g)
So, let's break down the steps to solve part (a) of the problem:
Step 1: Calculate the weight of the block.
Given that the mass of the block (m) is 3.5 kg and the acceleration due to gravity (g) is 9.8 m/s^2, you can calculate the weight (W) as follows:
W = m * g
W = 3.5 kg * 9.8 m/s^2
W ≈ 34.3 N
Step 2: Calculate the normal force.
The normal force (N) is equal to the weight of the block, so N = 34.3 N.
Step 3: Calculate the frictional force.
Given that the coefficient of kinetic friction (μ) is 0.25, you can now calculate the magnitude of the frictional force (Ff) using the formula:
Ff = μ * N
Ff = 0.25 * 34.3 N
Ff ≈ 8.575 N
Therefore, the magnitude of the frictional force on the block from the floor is approximately 8.575 N.
Now, let's move forward to part (b) of the problem, which is to calculate the magnitude of the block's acceleration.
To calculate the magnitude of the block's acceleration, you can use the following equation from Newton's second law:
net force (Fnet) = mass (m) * acceleration (a)
In this case, the net force acting on the block is the horizontal component of the applied force (F) minus the frictional force (Ff). Therefore, we can write:
Fnet = F * cos(θ) - Ff
where θ is the angle between the applied force and the horizontal direction, given as 35°.
Step 1: Calculate the horizontal component of the applied force.
Given that the magnitude of the applied force (F) is 17 N, you can calculate the horizontal component (Fhorizontal) as follows:
Fhorizontal = F * cos(θ)
Fhorizontal = 17 N * cos(35°)
Fhorizontal ≈ 13.923 N
Step 2: Calculate the net force.
Now that you have the horizontal component of the applied force (Fhorizontal) and the frictional force (Ff), you can calculate the net force (Fnet):
Fnet = Fhorizontal - Ff
Fnet = 13.923 N - 8.575 N
Fnet ≈ 5.348 N
Step 3: Calculate the acceleration.
Now, using Newton's second law, you can calculate the magnitude of the block's acceleration (a):
Fnet = m * a
5.348 N = 3.5 kg * a
a ≈ 1.528 m/s^2
Therefore, the magnitude of the block's acceleration is approximately 1.528 m/s^2.