Answer:
a = – 2.01m/s²
The magnitude of the acceleration is 2.01m/s²
Explanation:
Given m = 50kg, θ = 15°, μs = 0.5
The forces acting on the crate are
The weight W = mg
The horizontal force F
The static frictional force
The reaction at the surface of the cart R
Taking x-axis to be parallel to the surface of the cart and y-axis to be perpendicular to the surface.
Resolving the forces into components parallel and perpendicular to the surface of the cart we have that
Wx = mgSinθ = 50×9.8×Sin15° = +126.8N
Wy = mgCosθ = 50×9.8Cos15° = –473.3N
Rx = 0
Ry = R,
(Ff)x = Ff = –μs×R = –0.5R
(Ff)y = 0
Fx = maCosθ = 50a×Cos15° = –48.3a
Fy = maSinθ = 50a×Sin15° = –12.94a
By Newtown's first law the sum of all the forces is zero
Summing all x-components forces
Rx +Wx + Ffx + Fx = 0
0 + 126.8 –0.5R – 48.3a = 0
0.5R +48.3a = 126.8 .......(1)
Summing all y-component forces
Ry +Wy + Ffy + Fy = 0
R – 473.3 + 0 –12.94a = 0
R = 12.94a + 473.3
Substituting R in equation in (1)
0.5(12.94a + 473.3) + 48.3a = 126.8
6.47a + 236.7 + 48.3a = 126.8
54.77a = 126.8 – 236.7
54.77a = – 109.9
a = – 109.9/54.77 = – 2.01m/s²
a = – 2.01m/s²
