A 1500 kg block of granite is pulled up an incline that has an angle of inclination of 23 o with a constant speed of 2.03 m/s by a steam winch (see Figure). The coefficient of kinetic friction between the block and the incline is 0.29. How much power must be supplied by the winch?
To find the power supplied by the winch, we need to calculate the force of friction acting on the block and then multiply it by the velocity.
First, we need to determine the force of friction. The force of friction can be calculated using the equation:
Frictional force = coefficient of kinetic friction * normal force
The normal force is the force acting perpendicular to the incline, which can be calculated using the equation:
Normal force = mass * gravitational acceleration * cos(angle of inclination)
Given:
Mass (m) = 1500 kg
Angle of inclination (θ) = 23°
Coefficient of kinetic friction (μ) = 0.29
Velocity (v) = 2.03 m/s
Gravitational acceleration (g) = 9.8 m/s²
Calculating the normal force:
Normal force = 1500 kg * 9.8 m/s² * cos(23°)
Calculating the force of friction:
Frictional force = 0.29 * (1500 kg * 9.8 m/s² * cos(23°))
Next, we can calculate the work done by the frictional force:
Work = Frictional force * distance
The distance along the incline is not provided in the question. If the distance is not given, we cannot calculate the work and, consequently, the power supplied by the winch.
To calculate the power supplied by the winch, we need to understand the forces acting on the block on the incline. There are primarily two forces to consider: the force applied by the winch and the force of friction.
1. First, let's calculate the force of gravity pulling the block downward. The force of gravity can be calculated using the formula: F_gravity = mass × acceleration due to gravity.
F_gravity = 1500 kg × 9.8 m/s^2 ≈ 14,700 N
2. Next, let's determine the force of friction. The force of friction can be calculated using the formula: F_friction = coefficient of friction × normal force.
The normal force is the perpendicular force exerted by the incline on the block. It can be calculated using the formula: N = mass × gravity × cos(angle of inclination).
N = 1500 kg × 9.8 m/s^2 × cos(23 o) ≈ 13012 N
F_friction = 0.29 × 13012 N ≈ 3774.68 N
3. Since the block is moving at a constant speed, the applied force by the winch must exactly balance the force of friction. Therefore, the force applied by the winch is equal to the force of friction: F_applied = F_friction ≈ 3774.68 N.
4. Finally, we can calculate the power supplied by the winch using the formula: Power = Force × Velocity.
Power = F_applied × Velocity = 3774.68 N × 2.03 m/s ≈ 7665.64 Watts.
So, the winch must supply approximately 7665.64 Watts (or 7.67 kilowatts) of power.
normal force = 1500*9.81*cos 23
so
friction force = .29 *1500 *9.81*cos 23
gravity component down slope = m g sin 23 = 1500*9.81*sin 23
so
force required = 1500*9.81(.29cos 23 + sin 23)
work = force * distance
work/second = Joules/s = watts = 2.03*1500*9.81(.29cos 23 + sin 23)