A person wants to push a lamp (mass 7.2 kg) across the floor, for which the coefficient of friction is 0.23. Calculate the maximum height above the floor at which the person can push the lamp so that it slides rather than tips (Fig. 9-62).
To calculate the maximum height above the floor at which the lamp can be pushed so that it slides rather than tips, we need to consider the forces acting on the lamp and determine the conditions for sliding and tipping.
Let's first identify the forces acting on the lamp:
1. Weight (mg): The force exerted on the lamp due to gravity is given by the product of the mass (7.2 kg) and the acceleration due to gravity (9.8 m/s²). So the weight is W = 7.2 kg × 9.8 m/s².
2. Normal force (N): The support force exerted by the floor on the lamp, perpendicular to the surface, is equal in magnitude and opposite in direction to the weight (mg).
3. Friction force (f): The force opposing the motion of the lamp is the friction force. It can be calculated using the coefficient of friction (μ) and the normal force (N). The friction force can be expressed as f = μN.
The lamp will start to slide rather than tip when the friction force is equal to or greater than the force trying to tip it over. This tipping force is caused by the component of the weight acting perpendicular to the direction of motion.
The maximum height can be determined by the point at which the tipping torque is equal to the frictional torque:
Tipping torque = Frictional torque
Let's calculate that:
1. Tipping torque: The tipping torque is caused by the perpendicular component of weight and the height (h) at which the lamp is pushed. It can be calculated as Tipping torque = (mg)(h).
2. Frictional torque: The frictional torque is given by the product of the friction force (f) and the distance from the point of rotation (h). Since the lamp will pivot on one side, the distance is equal to the height (h). So the frictional torque is Frictional torque = fh.
Setting the tipping torque equal to the frictional torque, we have:
(mg)(h) = fh
Now we can solve for the maximum height (h):
(mg) = f
Substituting in the expressions for (mg) and f:
(7.2 kg × 9.8 m/s²) = (0.23)N
Since N is equal to the weight (mg), we can substitute that as well:
(7.2 kg × 9.8 m/s²) = (0.23)(7.2 kg × 9.8 m/s²)
Simplifying the equation:
7.2 kg × 9.8 m/s² = 0.23 × 7.2 kg × 9.8 m/s²
Solving for the maximum height:
h = (0.23 × 7.2 kg × 9.8 m/s²) / (7.2 kg × 9.8 m/s²)
Calculating the maximum height:
h ≈ 0.23 m
Therefore, the maximum height above the floor at which the person can push the lamp so that it slides rather than tips is approximately 0.23 meters.