A 1600 kg block of granite is pulled up an incline that has an angle of inclination θ = 20 ° with a constant speed of 1.21 m/s by a steam winch (see Figure). The coefficient of kinetic friction between the block and the incline is 0.09. How much power must be supplied by the winch?
M*g = 1600 * 9.8 = 15,680 N.
Fp = 15,680*sin20 = 5363 N.
Fn = 15,680*Cos20 = 14,734 N.
Fk = u*Fn = 0.09 * 14,734 = 1326 N.
Fw-Fp-Fk = M*a = M*0 = 0.
Fw = Fp+Fk = 5363 + 1326 = 6689 N. =
Force of the winch.
Power = Fw * V = 6689 * 1.21 = 8094 J/s.
= 8094 Watts.
To find the power supplied by the winch, we need to calculate the work done against friction and divide it by the time taken.
Step 1: Calculate the force of friction.
The force of friction can be calculated using the formula:
Frictional force = μ * normal force
where μ is the coefficient of kinetic friction.
The normal force can be calculated using the formula:
Normal force = mass * gravity * cos(θ)
where mass is the mass of the block and θ is the angle of inclination.
Plugging in the given values:
μ = 0.09
mass = 1600 kg
θ = 20°
Gravity can be approximated as 9.8 m/s^2.
Normal force = 1600 * 9.8 * cos(20°)
Step 2: Calculate the force required to move the block up the incline.
The force required to move the block up the incline is given by the formula:
Force required = mass * gravity * sin(θ)
Force required = 1600 * 9.8 * sin(20°)
Step 3: Calculate the work done against friction.
The work done against friction is given by the formula:
Work done against friction = force of friction * distance
The distance can be calculated using the formula:
Distance = velocity * time
Since the velocity is constant at 1.21 m/s, we can choose any time value.
We will assume a time value of 1 second for simplicity.
Distance = 1.21 m/s * 1 s
Step 4: Calculate the power.
Power = work done against friction / time
Now we can plug in the values calculated in the previous steps to find the power.
I will calculate the values in the next steps.
To find the power required by the winch, we first need to calculate the force required to overcome the friction and pull the block up the incline at a constant speed.
First, let's calculate the gravitational force acting on the block. The gravitational force can be calculated using the formula:
F_gravity = m * g
where m is the mass of the block and g is the acceleration due to gravity. In this case, the mass of the block is 1600 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
Substituting the values, we get:
F_gravity = 1600 kg * 9.8 m/s^2 = 15680 N
Next, let's calculate the force of kinetic friction acting on the block. The force of kinetic friction can be calculated using the formula:
F_friction = μ * F_normal
where μ is the coefficient of kinetic friction and F_normal is the normal force acting on the block.
The normal force can be calculated using the formula:
F_normal = m * g * cos(θ)
where θ is the angle of inclination. In this case, the angle of inclination is 20°.
Substituting the values, we get:
F_normal = 1600 kg * 9.8 m/s^2 * cos(20°) ≈ 15015 N
Now, substituting the value of the coefficient of kinetic friction (0.09) and the normal force (15015 N), we can calculate the force of kinetic friction:
F_friction = 0.09 * 15015 N ≈ 1351 N
We know that the block is being pulled up the incline at a constant speed, which means that the force applied by the winch is equal in magnitude and opposite in direction to the force of kinetic friction. Therefore, the force applied by the winch is 1351 N.
Finally, let's calculate the power required by the winch. Power can be calculated using the formula:
Power = Force * Velocity
Substituting the values of force (1351 N) and velocity (1.21 m/s), we get:
Power = 1351 N * 1.21 m/s ≈ 1634.71 W
Therefore, the power required by the winch is approximately 1634.71 Watts.