A 3.5 kg block is pushed along a horizontal floor by a force of magnitude 15 N at an angle 40° with the horizontal (Fig. 6-19). The coefficient of kinetic friction between the block and the floor is 0.25. Calculate the magnitudes of (i) the frictional force on the block from the floor
and (ii) the block’s acceleration.
M*g = 3.5 * 9.8 = 34.3 N. = Wt. of block = Normal force(Fn).
Fp = 15*Cos40 = 11.5 N. = Force parallel with the surface.
1. Fk = u*Fn = 0.25 * 34.3 = 8.58 N.
2. Fp-Fk = M*a. a = ?.
To calculate the magnitudes of the frictional force and the block's acceleration, we can follow these steps:
(i) Calculate the frictional force on the block from the floor:
1. Use the formula for frictional force: F_friction = coefficient of friction * normal force.
2. The normal force is the force exerted by the floor on the block perpendicular to the surface. In this case, it is equal to the weight of the block because there is no vertical acceleration.
Normal force = mass of the block * acceleration due to gravity.
3. Substitute the given values and calculate the normal force.
4. Apply the formula for frictional force and calculate its magnitude.
(ii) Calculate the block's acceleration:
1. Apply Newton's second law: F_net = mass * acceleration.
2. The net force acting on the block is the horizontal component of the applied force minus the frictional force.
F_net = F_applied * cos(angle) - F_friction.
3. Substitute the given values and calculated frictional force.
4. Solve for acceleration by rearranging the formula.
Let's proceed with the calculations.
Given:
Mass of the block (m) = 3.5 kg
Applied force (F_applied) = 15 N
Angle (θ) = 40°
Coefficient of kinetic friction (μ_kinetic) = 0.25
Acceleration due to gravity (g) = 9.8 m/s²
(i) Calculating the frictional force:
1. Normal force (N) = m * g
Normal force = 3.5 kg * 9.8 m/s² = 34.3 N
2. Frictional force (F_friction) = μ_kinetic * N
F_friction = 0.25 * 34.3 N = 8.575 N
Therefore, the magnitude of the frictional force acting on the block from the floor is 8.575 N.
(ii) Calculating the block's acceleration:
1. Net force in the horizontal direction (F_net) = F_applied * cos(θ) - F_friction
F_net = 15 N * cos(40°) - 8.575 N ≈ 6.616 N
2. Applying Newton's second law: F_net = m * a
6.616 N = 3.5 kg * a
Solving for acceleration (a):
a = 6.616 N / 3.5 kg ≈ 1.89 m/s²
Therefore, the magnitude of the block's acceleration is approximately 1.89 m/s².