A 350 piano slides 3.5 down a 33 incline and is kept from accelerating by a man who is pushing back on it parallel to the incline (see the figure ). The effective coefficient of kinetic friction is 0.30. Calculate the force exerted by the man.
To calculate the force exerted by the man, we need to consider the forces acting on the piano and the forces causing them.
1. Force of gravity (Fg): This force acts vertically downwards and can be calculated using the formula Fg = m * g, where m is the mass and g is the acceleration due to gravity (approximated to 9.8 m/s²).
Given the mass of the piano (m) as 350 kg, the force of gravity can be calculated as follows:
Fg = 350 kg * 9.8 m/s²
2. Force parallel to the incline (Fp): This is the force exerted by the man to keep the piano from accelerating down the incline. It opposes the force of gravity.
3. Force of friction (Ff): This is the force that opposes the motion of the piano. It is given by the formula Ff = μ * N, where μ is the coefficient of kinetic friction and N is the normal force.
The normal force (N) can be calculated as follows:
N = m * g * cos(θ), where θ is the angle of the incline (in this case, 33°).
The force of friction can then be calculated:
Ff = μ * N
Since the force parallel to the incline (Fp) and the force of friction (Ff) are in opposite directions, we can write:
Fp - Ff = 0
Substituting the expressions for Ff and N, and rearranging the equation, we can solve for Fp:
Fp = μ * m * g * cos(θ)
Now, let's calculate the force exerted by the man:
Fg = 350 kg * 9.8 m/s² (approximating to 3430 N)
N = 350 kg * 9.8 m/s² * cos(33°)
Ff = 0.30 * N
Fp = 0.30 * 350 kg * 9.8 m/s² * cos(33°)
To calculate the force exerted by the man, we need to analyze the forces acting on the piano.
1. The force of gravity (weight): The weight of the piano can be calculated using the formula W = mg, where m is the mass of the piano and g is the acceleration due to gravity (approximately 9.8 m/s²). Since the weight acts vertically downward, it can be represented as W = 350 kg * 9.8 m/s² = 3430 N.
2. The force due to the incline: The force due to the incline can be calculated using the formula F_incline = mg * sin(θ), where θ is the angle of the incline. In this case, the angle is given as 33°. Therefore, F_incline = 350 kg * 9.8 m/s² * sin(33°).
3. The force of friction: The force of friction can be calculated using the formula F_friction = μ * N, where μ is the coefficient of friction and N is the normal force. The normal force can be calculated using the formula N = mg * cos(θ), where θ is the angle of the incline. Therefore, N = 350 kg * 9.8 m/s² * cos(33°). Once we have the normal force, the force of friction can be calculated as F_friction = 0.30 * N.
4. The force exerted by the man: The force exerted by the man should be equal in magnitude and opposite in direction to the force due to the incline plus the force of friction. Therefore, the force exerted by the man can be calculated as F_man = F_incline + F_friction.
Now, you can plug in the values and calculate the force exerted by the man:
F_incline = 350 kg * 9.8 m/s² * sin(33°)
N = 350 kg * 9.8 m/s² * cos(33°)
F_friction = 0.30 * N
F_man = F_incline + F_friction.
(Note: Make sure to convert the angle from degrees to radians when using the trigonometric functions in your calculations.)
Calculating these values will give you the force exerted by the man.