A basket of flowers weighing 2kg is placed on a flat grassy slope. The coefficient of static friction between the basket and the slope is 0.4, and the basket is on the point of slipping down the slope.
Draw a force diagram to indicate forces (including friction) that act on the basket.
Write down vector expressions for the forces and the equilibrium condition. Hence determine the angle that the slope makes with the horizontal.
Really not sure where to even start
M*g = 2 * 9.8 = 19.6 N. = Wt. of flowers.
Fp = 19.6*sin A. = Force parallel with incline.
Fn = 19.6*Cos A. = Normal force.
Fs = u*Fn = 0.4*19.6*Cos A = 7.84*Cos A. = Force of static friction.
Fp-Fs = M*a.
19.6*sin A-7.84*Cos A = M*0 = 0,
19.6*sin A = 7.84Cos A,
19.6*sin A/Cos A = 7.84,
sin A/Cos A/Cos A = 7.84/19.6 = 0.4,
sin A/Cos A = Tan A,
Tan A = 0.4, A = 21.8o.
To solve this problem, we can start by drawing a force diagram to represent the forces acting on the basket of flowers.
Step 1: Draw the basket of flowers as a rectangle shape on the slope.
Step 2: Identify the relevant forces acting on the basket:
1. Weight (W): This is the force of gravity acting on the basket, directed vertically downward. Its magnitude is given by the formula W = mg, where m is the mass of the basket and g is the acceleration due to gravity.
2. Normal force (N): This is the force exerted by the slope on the basket perpendicular to the surface of the slope. It acts in the direction perpendicular to the slope and counterbalances the weight of the basket. Its magnitude is equal to the weight of the basket.
3. Friction force (f): This is the force of static friction that prevents the basket from slipping down the slope. It acts parallel to the surface of the slope and opposes the motion. Its magnitude can be calculated using the formula f = μN, where μ is the coefficient of static friction and N is the normal force.
Now, we can write out the vector expressions for these forces:
Weight (W) = mg (directed vertically downward)
Normal force (N) = -mg (directed perpendicular to the slope and upward)
Friction force (f) = -μN (directed parallel to the slope and opposite to the direction of potential motion)
Next, we can determine the equilibrium condition for the basket. In equilibrium, the net force acting on the basket must be zero.
Net force = W + N + f = 0
Now, let's substitute the vector expressions:
mg - mg - μN = 0
Since N = mg, the equation simplifies to:
mg - mg - μ(mg) = 0
Simplifying further:
(1 - 1 - μ)mg = 0
Since the basket is on the point of slipping, μ = coefficient of static friction = 0.4. Substituting:
(1 - 1 - 0.4)mg = 0
-0.4mg = 0
Dividing both sides by -0.4g:
m = 0
This means that the mass of the basket cancels out, so it doesn't affect the equilibrium condition, and the slope angle can be determined regardless of the mass.
Finally, we can solve for the slope angle:
The slope angle (θ) can be found using the relationship:
tan(θ) = μ = 0.4
Taking the inverse tangent of both sides (using a calculator or trigonometry table):
θ = arctan(0.4)
Using a calculator, we find that the slope angle θ is approximately 21.8 degrees.
Therefore, the slope makes an angle of approximately 21.8 degrees with the horizontal.