A sphere hangs suspended by a light string, resting against a vertical wall. The sphere has a mass of 2 kg and the string makes an 80° angle with the horizontal. What is the force from the wall against the ball?

Alright, guys. Let's check this one out. So we get the spear that's hanging against a vertical wall like this. If you've ever played a game of tetherball, you know, it's basically as if, like, this ball was resting up against the pole or something like that. Alright. So we're told the angle that it makes to the horizontal, that 80 degrees, and what we wanna do is we wanna figure out the force that the wall is applying against the ball. So to do that, we wanna figure out one of the forces. We wanna draw a free body diagram. So let's go ahead and get started. So we've got our ball or like this, and then we've got the m g force that's gonna be downwards. Right? Then we also have an applied force or tensions. There's no applied forces. There's nothing pushing or pulling this, but we do have the string here. So we do have this tension force that's gonna act at some angle. We actually know what the angle is. We know this is 80 degrees. Alright. So then we look for any, contact forces, and we do actually have 2 surfaces that are touching. So, basically, this point of the sphere is resting up against the surface because the tension is basically wanna pull it against the wall. So the tension actually exerts a surface push or makes the sphere push against the wall. So the normal force is a reaction to that surface push and it's always perpendicular to the surface. So it's actually gonna go out like this. So you do have a normal force except it doesn't point straight up. It actually points to the left. This is actually what we're trying to solve in this problem here. This is the force from the wall against the ball because of that surface push. Right? So, really, what we're trying to do is solve the normal. So now what we do is we decompose any two dimensional forces that we have. Our normal points to the left are mg points down, but we have this tension force that points at some two dimensional angles. So we separate into its components tx and ty. And if I have the magnitude and direction, I can calculate them. Unfortunately, I only have the direction. I don't have what t is equal to but I know that t x is gonna be t cosine theta and I know that t y is gonna be t times the sine of theta. Right? So now what I can do is I basically just go to the next step, which is I'm gonna I'm gonna write out some of our forces in x and y. Right? Or I'm gonna write f equals m a because I wanna figure out what that normal force is. So in the x axis, where I've got is f equals ma, and in the y axis, I've got f f equals ma So let's look at the accelerations. Well, if you think about this, if this sphere is suspended against the wall from this cable and it's resting up against the wall, then what that means is to say this is a 2 dimensional equilibrium problem. Right? All the forces have to cancel out. This is a 2 d equilibrium problem. All the forces have to cancel out in the x and y so that there's no acceleration in either direction. That's what equilibrium means. So if we do this, we're gonna have to expand it all over our forces. We're gonna have to choose a direction of positive. So if we wanna figure out the normal force, that's really what we're trying to solve for, and it points to the left, then we should actually choose the left direction to be positive and then upwards to be positive. That way, our normal force is actually gonna be positive. Right? So, we expand all over forces. We've really only got the n's. We've really got the normal force and the t x. So my normal is actually against or along the direction of positive, and my t x is against the direction of positive. So that's gonna be 0. Alright? So that's gonna be negative t x equals 0. Alright. So, basically, what I've got is n is equal to tx. So I wanna figure out this normal force, but I need to figure out what this tx is equal to. Remember, we can actually expand this because we know that tx is t cosine theta. So really, I've got normal is equal to t times the cosine of 80 degrees. Alright. So if I wanna figure out the normal force, I need to figure out all the other variables in this equation. Unfortunately, I don't know what the magnitude of that tension is. Right? What's that t value? We didn't know that. So what happens is if we get stuck, we're just gonna move to the y axis. See if we can find we'll see if we can solve the tension there. So all of our forces in the y axis are gonna be ty and then we're gonna have mg downwards So this is gonna be mg is equal to 0. So we know those two things have to equal each other So remember we can expand this t y because we know that t y is gonna be t times the sine of 80. So this is gonna be mg. Alright, and we actually know everything about this equation except for T so we can go ahead and solve so this T is really equal to mg divided by the sine of 80, so it's really just 2 times 9.8 divided by the sine of 80. If you go ahead and work this out, you're gonna get 19.9 Newtons. So this 19.9 means we can just plug it back into our original equation in the x axis. So the normal is just equal to 19.9 times the cosine of 80 degrees. And if you work this out in your calculator, you're gonna get a normal force of 3.5 newtons. So we look through our answer choices, and that's answer choice c. Alright, guys. So that's it for this one.

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A sphere hangs suspended by a light string, resting against a vertical wall. The sphere has a mass of 2 kg and the string makes an 80° angle with the horizontal. What is the force from the wall against the ball?

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