the coefficients of static and kinetic friction for stone on wood are, respectively, 0.5 and 0.4. if a 150 kg stone statue is pushed with just enough force to start moving across a wooden floor and the same force continues to act afterward. find the statue’s acceleration.
force=ma where
force=mg*(.5-.4)
a= .1g
check that
To find the acceleration of the statue, we need to consider the forces acting on it. The force required to overcome static friction initially, and then the force of kinetic friction once it starts moving.
1. Calculating the force to overcome static friction:
The force required to overcome static friction is given by the formula:
F_static = μ_static * N
where μ_static is the coefficient of static friction and N is the normal force.
The normal force can be calculated as:
N = m * g
where m is the mass of the statue (150 kg) and g is the acceleration due to gravity (9.8 m/s^2).
N = 150 kg * 9.8 m/s^2 = 1470 N
Now, we can calculate the force to overcome static friction:
F_static = 0.5 * 1470 N = 735 N
2. Calculating the force of kinetic friction:
Once the statue starts moving, the force of friction changes from static friction to kinetic friction. The force of kinetic friction is given by the formula:
F_kinetic = μ_kinetic * N
where μ_kinetic is the coefficient of kinetic friction.
F_kinetic = 0.4 * 1470 N = 588 N
3. Calculating the acceleration:
Now that we have the forces acting on the statue, we can calculate the net force:
Net Force = Force_applied - Force_friction
Force_applied = 735 N (as it is the same force used to start moving the statue)
Net Force = 735 N - 588 N = 147 N
Using Newton's second law (F = m * a), we can calculate the acceleration:
a = Net Force / m
a = 147 N / 150 kg = 0.98 m/s^2
Therefore, the statue's acceleration is 0.98 m/s^2.
To find the statue's acceleration, we will use Newton's second law of motion:
\(F_{\text{net}} = m \cdot a\),
where \(F_{\text{net}}\) is the net force acting on the object, \(m\) is the mass of the object, and \(a\) is the acceleration.
First, let's determine the net force acting on the stone statue. The force pushing the statue is equal to the force of friction, which can be calculated using the formula:
\(f = \mu \cdot N\),
where \(f\) is the force of friction, \(\mu\) is the coefficient of friction, and \(N\) is the normal force.
Since the statue is on a horizontal wooden floor, the normal force (\(N\)) is equal to the weight of the statue (\(mg\)), where \(g\) is the acceleration due to gravity (approximately \(9.8 \, m/s^2\)).
Therefore, the force of friction can be calculated as:
\(f = \mu \cdot N = \mu \cdot mg\).
Substituting the given values of \(\mu\) and \(m\):
\(f = 0.5 \cdot 150 \, \text{kg} \cdot 9.8 \, m/s^2\).
Next, since the statue is being pushed with just enough force to start moving, the applied force equals the force of static friction (\(F_{\text{app}} = f_{\text{static}}\)). Once the statue starts moving, the applied force equals the force of kinetic friction (\(F_{\text{app}} = f_{\text{kinetic}}\)). In this case, since the same force continues to act afterward, the applied force remains the same.
To determine the acceleration, we need to find the net force acting on the statue. Since the applied force (\(F_{\text{app}}\)) is equal to the force of friction (\(f = f_{\text{static}} = f_{\text{kinetic}}\)), the net force is:
\(F_{\text{net}} = F_{\text{app}} - f = F_{\text{app}} - \mu \cdot mg\).
Finally, we can use Newton's second law (\(F_{\text{net}} = m \cdot a\)) to find the acceleration:
\(m \cdot a = F_{\text{net}} \Rightarrow a = \frac{F_{\text{net}}}{m}\).
Substituting the given values, we get:
\(a = \frac{F_{\text{app}} - \mu \cdot mg}{m}\).
Calculating the values:
\(a = \frac{(0.5 \cdot 150 \, \text{kg} \cdot 9.8 \, m/s^2) - (0.5 \cdot 150 \, \text{kg} \cdot 9.8 \, m/s^2)}{150 \, \text{kg}}\).
After performing the calculations, we find that the acceleration of the stone statue is 0 \(m/s^2\).