A 400-gram package lying on a horizontal surface is attached to a horizontal string which passes over a smooth pulley. When a mass of 200 grams is attached to the other end of the string, the package is on the point of moving. Find the coefficient of friction.
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To find the coefficient of friction, we need to consider the forces acting on the system.
1. Weight of the 400-gram package: Given it has a mass of 400 grams (0.4 kg) and acceleration due to gravity is approximately 9.8 m/s², the weight force acting on the package is:
Weight1 = mass1 × acceleration due to gravity
= 0.4 kg × 9.8 m/s²
= 3.92 N
2. Tension in the string: The tension force in the string is the same on both sides of the pulley. When a mass of 200 grams (0.2 kg) is attached to the other end, the tension force in the string is:
Tension = mass2 × acceleration due to gravity
= 0.2 kg × 9.8 m/s²
= 1.96 N
3. Friction force: The package is on the point of moving, which means the static friction is at its maximum. The friction force is given by:
Friction = coefficient of friction × Normal force
4. Normal force: The normal force is equal to the weight force of the package, since the surface is horizontal and there is no vertical acceleration.
Normal force = Weight1
= 3.92 N
Now, since the system is on the point of moving, the tension force of the string is equal to the friction force:
Tension = Friction
Substituting the values we have:
1.96 N = coefficient of friction × 3.92 N
To find the coefficient of friction, divide both sides of the equation by 3.92 N:
coefficient of friction = 1.96 N / 3.92 N
Simplifying:
coefficient of friction ≈ 0.5
Therefore, the coefficient of friction is approximately 0.5.
To find the coefficient of friction, we need to use the concept of equilibrium. The package is on the point of moving, which means that the force of friction is equal to the force exerted by the hanging mass.
Let's break down the forces acting on the package:
1. The force of gravity on the package (weight): W = mg, where m is the mass and g is the acceleration due to gravity.
2. The normal force exerted by the table on the package: N = mg, because the package is on a horizontal surface and there is no vertical acceleration.
3. The force of friction: Ff = μN, where μ is the coefficient of friction. This force opposes the impending motion of the package.
4. The force exerted by the hanging mass: Fh = mass × g, since mass is the hanging mass and g is the acceleration due to gravity.
Since the package is on the point of moving, we can set up the equation for equilibrium:
Fh = Ff
mass × g = μN
Substituting the values, we have:
0.2 kg × 9.8 m/s^2 = μ × (0.4 kg × 9.8 m/s^2)
Simplifying the equation:
0.2 × 9.8 = 0.4 × 9.8 × μ
1.96 = 3.92 × μ
Dividing both sides by 3.92:
μ = 1.96 / 3.92
μ = 0.5
Therefore, the coefficient of friction is 0.5.
M1*g = 0.4 * 9.8 = 3.92 N. = Wt. of pkg.= Normal force(Fn).
Fp = 3.92*sin 0 = 0 = Force parallel to the surface.
Fap = M2*g = 0.2 * 9.8 = 1.96 N. = Force applied.
Fs = us*Fn. = 3.92us.
Fap-Fp-Fs = M*a.
1.96-0-3.92us = M*0 = 0, us = 0.50.