A 2.8 kg block is pushed along a horizontal floor by a force F of magnitude 19 N at an angle 𝜃 = 35° with the horizontal (Fig. 6-49). The coefficient of kinetic friction between the block and floor is 0.25. (The block initially starts at rest.)Figure 6-49(a) Calculate the magnitude of the frictional force on the block from the floor. N(b) Calculate the magnitude of the acceleration of the block. m/s2
Question
A 2.8 kg block is pushed along a horizontal floor by a force F of magnitude 19 N at an angle 𝜃 = 35° with the horizontal (Fig. 6-49). The coefficient of kinetic friction between the block and floor is 0.25. (The block initially starts at rest.)Figure 6-49(a) Calculate the magnitude of the frictional force on the block from the floor. N(b) Calculate the magnitude of the acceleration of the block. m/s2
Solution
(a) The frictional force (F_friction) can be calculated using the formula:
F_friction = μk * F_normal
where μk is the coefficient of kinetic friction and F_normal is the normal force.
The normal force in this case is the weight of the block minus the vertical component of the applied force. The weight of the block (F_weight) is its mass (m) times the acceleration due to gravity (g), and the vertical component of the applied force (F_vertical) is F * sin(θ).
So, F_normal = F_weight - F_vertical = mg - Fsin(θ) = 2.8 kg * 9.8 m/s² - 19 N * sin(35°) = 27.44 N - 10.96 N = 16.48 N
Then, we can substitute F_normal into the formula for F_friction:
F_friction = μk * F_normal = 0.25 * 16.48 N = 4.12 N
(b) The acceleration of the block (a) can be calculated using Newton's second law (F = ma), where F is the net force and m is the mass of the block.
The net force is the horizontal component of the applied force (F_horizontal) minus the frictional force (F_friction). F_horizontal is F * cos(θ).
So, F_net = F_horizontal - F_friction = F * cos(θ) - F_friction = 19 N * cos(35°) - 4.12 N = 15.56 N - 4.12 N = 11.44 N
Then, we can substitute F_net into the formula for a:
a = F_net / m = 11.44 N / 2.8 kg = 4.09 m/s²
So, the magnitude of the frictional force on the block from the floor is 4.12 N and the magnitude of the acceleration of the block is 4.09 m/s².
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